Looking for a Tutor Near You?

Post Learning Requirement »
x

Choose Country Code

x

Direction

x

Ask a Question

x

x
x
For better view click here
x
Hire a Tutor
DISCOVER

Analytical Geometry

Loading...

Published in: Mathematics
8,421 Views

Direction cosines and ratios. Angle between two line – Equation of a plane – Equation of a straight line – Co – planar lines – shortest distance between skew lines – sphere – Tangent plane – Plane section of a sphere – Orthogonal spheres.

C K / Chennai

11 years of teaching experience

Qualification: M.Sc., M.Phil.,

Teaches: Mathematics

Contact this Tutor
  1. UNIT - 11 Three Dimensional Analytical Geometry: Direction cosines and ratios. Angle between two line Equation of a plane Equation of a straight line Co — planar lines shortest distance between skew lines sphere Tangent plane Plane section of a sphere Orthogonal spheres. Introduction: Let x ox, y oy and z oz be three mutually perpendicular lines in space, these three lines are called as x axis, y- axis and z-axis respectively (co —ordinate axes). The period of intersection of the axes is the origin o. The positive parts of the co-ordinates axes, namely, ox, oy, oz should form a right- handed system. Any point P in space is uniquely known by its distances from the planes yoz, zox and xoy Parallel to ox, oy and oz respectively. They are respectively called its x, y and z co- ordinates. The above planes are known as coordinate planes, yz- plane, zx plane and zy planes respectively. Note: Using the above definitions of co-ordinates, we see that the x co-ordinate of any point in the yz plane will be zero, the y co-ordinate of any point in thezx plane will be zero and the z- coordinate of any point in the xy — plane will be zero. The point A lies on the x-axis. Hence the co-ordinates of A will be (x,0,0). Similarly, the co-ordinates of B and C will be (0,y,0) and (0, 0, z) respectively. Result: 1. 2. 3. The distance between the two point P Y], ) and Q (x2, )'2, z2 ) is given by if P ) and Q (x2, )'2, z2 ) are two points, then the coordinates of the point p which divides PQ externally in the ratio : is ml x2 + 1112 xl ml y 2 +1712 y 1 ml z 2 + 1112 z 1 1711 -k 1712 1711 + 1711 -k 1712 The co-ordinates of the point, which divides PQ externally in the ration ntl.• is nil x2 — 1112 x 1 ml Y2 — '712 z 1 ml — 2 1
  2. 4. 5. The Centroid of the triangle whose vertices are P ) and Q (x2, Y'2, z2 ) and R G(x, y, z) 3 3 3 The centroid of the tetrahedral whose vertices are P ) and Q (x2, Y'2, z2 ), R( x3, Y3, z3) and S ( x4, Y'4, z4) is 4 4 4 Direction Cosines (D,C) : Let P (x, y, z ) be any point in space and let OP = r, if M is the foot of the perpendicular from P on the x axis, then 0M = x Let OP make angle with the positive x,y and z axes. Then D.C'S of OP are (Cosu,Cosß, Cosy) and usually denoted as (l, m, n). From the triangle OMP, 0M Cos u Similarly, Cos ß x — from(l) & Coo = :. Direction cosines of the line OP are Note: Let (l,m,n) be the D.C's of a line AB consider the line OP parallel to AB, then the D.C'S of OP are also Let OP = r, then the D.C. 'S of OP are(X/r, Y/r, Z/r Hence 1 = Yr' m = Y/r n — Now op2 = r 2 = r , by distance between two point ( Yr) 2 + (Y/r)2 + ( / ) 12 + + n = Thus the sum of the squares of D.C'S of line is equal to 1. Direction Ratios (D.R): Direction ratios of a straight line are a set of numbers a, b, c that are proportional to its Direction cosines. Let (a, b, c) be the D.R'S and (1, m, n ) be the D.C.'S of a line. Then 2
  3. k (say) But + n 1 k D.C.'S are a I 2 m a k' 1 a n b k' 2 n b 2 2 Note : 1. 2. 12m3+n = 1 but a +b2+c2# 1 The D.R. 'S of two parallel lines are proportional. Important Results: 1. 2. 3. Direction rations of the line joining two points P Y], ) and Q (x2, Y2, ), are (x2-x/, Y2-Y1, Q-ZI). Angle between two lines (a) if 0 is the angle between two lines whose direction cosines are (11, ml, 111) (b) and (12, 1712, 112) respectively then If the two lines are perpendicular, then 0 = 900 or Cos 0 = 0 if the two liens are parallel, then 11 = 12, if 0 is the angle between two lines whose direction ratios are (al, bl,C1) and respectively then cos 0 a la 2 -k CIC2 2 if the two lines are perpendicular, then 0 = 900 or Cos 0 = 0 i.e., al a24- blb2 + CIC2 if the two lines are parallel , then a 2 2 Projection of a line segment on a given line. Projection of the segment joining P(XI, Y], z]) and Q(X2, Y'2, z2) on a line AB whose D.C'S are (l, m, n) is Projection of PQ on AB. Xl) + m (Y2-Y1) + n (Q-ZI). 3
  4. Example 1: A line makes an angle of 450 with ox and 600 with oy. What angle does it makes with oz? Solve: Given 450, ß = 600 Let l, m, n be the D.C. 'S of the line OP. then 450 = 1/0 m=Cosß 60 - n = Cos Y But 12 +m +n = 1 2 1 2 1 2 - 1/2 - 1/4 = 1/4 — ± 1/2 ., = Cosy=lh Y = cos -1 1/2 = 600 Hence the line OP makes an angle 600 with oz Example 2: Find the D.C.'S ofthejoin of P (2 3, 5) and Q 3, 2) Solve Direction Ratios of the line joining P & Q is (x2 x], Y], z2-z1 )i.e., (-3, 0, D.C. 'S of the line are -3) 3 9+0+9 3 9+0+9 3 i.e., 1 i.e., Example — 3: 3 1 Show that the points A (4,0,5) , B(2, 2, 3) and C (1, 3, 2) are collinear. 4
  5. Solve: D.R.'S of AB are (2-4, 2-0, 3-5 ) i.e., (-2, 2, -2) D.R. 'S of BC are 1, -1) From (1) & (2) 2 1 1 1 ...(2) Thus the D.R. 'S of AB are proportional to the D.R. 'S of AC, :. AB and BC are parallel :. the points A, B, C are collinear Example 4: Find the angle between the lines whose direction ratios are (1 3 , , -1) and 2, 5) Solve: If 0 is the angle between the lines, then 1(-1 cos0 1+32+1 1+22 +52 = 900 SO O Example 5 : Find the angle between the two diagonals of a cube. Solve: Let the length of each side be 'a'. OP and RB is a pair of diagonals, where D.R.'S of the diagonal OP (a, a a) 1 Hence D.C'S of OP are D.R'S of the diagonal RB (a, a, -a) 1 Hence D.C'S of RB are G' G' 5
  6. :. the angle 0 between the two diagonals is given by 1 COS O = 111 333 O = COS Example — 6: 1 1 1 1 3 1 3 If u, ß, y, ö are the angles made by the diagonals of a cube with a line, then show that Cos u + Cos2ß + Cos y + Cos ö Solve: Let the length of each side be 1. Then the ends of the 4 diagonals are So the D.R'S of the diagonals are (1,1,1) ; ; ; (1,1,-1) If the D.C'S of the line are (l, m, n) then Cos , Cos ß= cos 9 , Cos ö= Squaring and simplifying cos u + cos2ß + cos 7+ cos Example — 7: 4 3 Find the projection of the line joining P(2,4,5) and R (3,6,7) on the line whose direction ratios are 2,3,2. Solve: D.R'S of the line are 2 D.C'S of the line are 17 3 17 2 . The projection of the line PQ on the line = I (x2 2 3 2 — (3-2) + (7-5) 2 3 2 (2) 6 Xl) + m (yr YD + n (z2-z1)
  7. Example — 8 : Find the projection of the line joining P(4, 2, 3) and Q (1, makes 300, 1200 and 900 with coordinate axes. Solve: The D.C'S ofthe line are cos 300, cos 1200, cos 900 The projection of PQ on the line (1-4) 1/2 (-3-2) +0 2 —(5 30) 2 Example 9: -3, 4) on a line which Find the projection of AB on CD, if A(l, 2, 2), B O, 2), CO, 1,2) and D (0, 5, 6). Solve: D.C's of CD are (-1, 4, 4) D.R.S of CD are 33' 33 :. Projection of PQ on CD 1 4 4 33 4 33 3) 33 2 33 -10 33 10 33 Example 10: 8 33 33 4 33 Show that the pair of lines whose direction cosines are given by 21-m+2n mn+nl+lm = 0 are perpendicular. Solve: To solve for l,m,n eliminate m 21-m+2n = 0 or m = 21 + 2n Substituting in mn+nl+lm = 0 (21+2n) n + ni + 1 (21+2n) = O 212 +5n1+2n2 O (1+2n) (21 + n) = O I = -2n or I = -Y2 7
  8. Case : 1: If 1 = -2n 21-m+2n = O -4n-m+2n = 0 m = -2n .. I :m:n = -2n : 2n ; n so (-2,-2, 1) are the D.R'S of a line Case — 2: n If 1 = 2 21 m + 2n = 0 -n — m +2n = 0 m=n :.l:m:n So (-1, 2, 2) are the D.R'S of another line n 2 ...(2) From (1) & (2) we see that the lines are perpendicular, since i.e., (2) + (-2) (2) + (1) (2) = O Exercise —I: 1. 2. 3. 4. Find the direction cosines of the line which makes 450 ox, 600 with oy and 1200 with oz If a line makes an angle 1200 with x make with the z- axis? 1 [Ans : axis, 450 wit y axis, what angle does it [Ans: 600 or 1200] Find the direction cosines of the line whose direction ratios are 3 Ans: (i) 50 2 (ii) — 14 A line is equally inclined with the x,y, z axis. Find the angles [ Ans: Cos 8 4 50 14 1 5 50 3 14
  9. 5. 6. 7. 8. 9. Find the direction ratios and direction cosines of the line joining the points (i) (1, 2, -1) and (2, 1, 3) (ii) ( and (2, 1, 1) (iii) (5, -2,7) and 1 [Ans ( 1, 30' 30' 30 121 [Ans: , Jö'Jö'Jö 3 [Ans: (3, -5,2) ; 38' 38' 38 Show that the following points A,B, C are Collinear (ii) A(3, -1,1), 5, -4, 2) , CUI, -13, 5) if (2, 1, -3) and (1, -3, 2) are D.R'S of two lines. Find the acute angle between them. [Ans : 1200] Prove that the line whose D.C'S are given by 31+m+2n= 0 and 41m—3mn-nl = 0 are perpendicular. Find the angle between the line whose D.C.'S are given by the equations 1+m+n=0, 21m-mn+21n = 0. [Ans : 600] 10. Show that the line whose D.C'S are given by 1+m+n = 0 and a21 +b m + c n = 0 are (i) perpendicular if a+b+c = 0 and (ii) parallel if l/a+l/b+l/c = 0. 11. Show that the lines whose D.C'S are given by a21 + b2m + c n = 0 and 1m + mn + nl = 0 are (i) perpendicular if l/a2 + 1/b2 + 1/c2 = 0 and (ii) parallel if a ± b ± c = 0 12. If two pair of opposite edges of a tetrahedron are perpendicular, show that the third pair is also perpendicular. 13. If a line makes angle with the co-ordinate axes, prove that Sin u + Sin ß + Sin y = 2. 14. If a line makes angle ö with the four diagonals of a cube. Prove that Sin u + Sin ß + Sin y + Sin ö = 8/3 15. Find the angle between the lines whose direction cosines are 1 4' 1 (ii) 2 1 4 1 [Ans: 600] [Ans: 450] 2 2 9
  10. 2 (iii) TI'FI'FI' 3 14 1 14 2 14 [Ans: 900] 16. Find the angle between the lines whose direction ratios are -1 cos 203 (ii) 4, 0-1, -0-1; 4, 17. Find the projection of the line joining the point (3, 2, 1) and (5, joining the points (1,-2, 6) and (-3, -1, 8) [Ans.' n /3] -3, 4) on the line [Ans : 18. If A,B,C,D are the points (1,0,5),(-1,2,4),(3,4,5) and (4,6,3) respectively. Find the projection of AB on CD. [Ans.• 4/3] 19. Find the Projection of AB on CD if a B (-1, 0,2), C and D The Plane: In a surface, if every point of the straight line joining any two points on it lies on it, the surface is a plane surface. General form of the Equation of a plane: The general equation of the first degree in x,y,z namely ax+by+cz+d = 0 always represent a plan. Note: a, b, c are D.R'S of the normal line to the plane ax+by+cz+d = 0. Cor-l: Equation of the plane passing through a given point Y], z]) and having a normal whose D.R'S are a,b,c is given by a(x-xl) + b (y-YI) + c (z-ZI) = 0 Cor- 2: Intercept form of a plane which cuts off intercepts a, b, c on the coordinate axes is given by — + — + 1 Cor- 3: xCos u +yCos ß+zCos y = p represent a plane at a distance p fro, the origin,u, ß, y being the direction angles of a normal to the plane. Cor — 4: The equation of the planes passing through the three points (x2, Y'2, z2) and (x3, Y'3, z3) is given by 10
  11. x Cor — 5:Length of the perpendicular p, from the origin on the plane ax+by+cz+d= 0 is given by 2 Cor — 6:Length of the perpendicular p, from the point Y], z]) on the plane ax+by+cz+d= 0 is given by Results: 1. LIXI -k by 1 + CZI + d a + b2+c 2 The angle between any two planes is equal to the angle between the respective normals, let the two planes be a IX+b1y+CIZ+d1 = O and a2x+b2y+c2z+d2 = 0, then, COS O = CIA2 -k blb2 -k CIC2 a 12 + b 12 + Cl a 22 + b22 + c 2 2 If the two planes are perpendicular, their normals are also perpendicular. Hence the two planes are perpendicular if alti2+b1b2+cß2 = O If the two planes are parallel, their normals are also parallel, Thus the condition for parallelism is a 1 a 2 2 Plane through the intersection of two given planes: Let the two planes be PI : a IX + b 1 y+ c IZ + d 1 = O and P 2 : C12X + b2Y + c2z + = O Then the equation of the plane through the intersection of two planes is given by PI + XP2=O Example 1: Find the D.C'S of the normal to the plane 2x+2y-z = 9 11
  12. Solve: The coefficients of x,y,z, namely (2,2,-1) are the D.R. 'S of the normal to the plane 221 The direction cosines are 3'3'3 Example — 2: Find the equation of the plane through (1,2,3) and parallel to 3x+4y-5z Solve: If two planes are parallel they differ by the constant term. Equation of the required plane is parallel to the plane 3x+4y-5z = 0 is given by 3x+4y-5z+k = O It is also given that the required plane passes through (1,2,3) + 4(2) 5(3) +k=O . The required plane is 3x+4y-5z+4 = 0. Example 3: Find the equation of the plane through (1,-2, 2) and (-3,1,-2) and perpendicular to the plane 2x+y-z+6 = 0 Solve: Equation of the required plane which passes through (1 ,-2,2) is x-1) + 6 (y +2) + c(z-2) = O it also passes through (-3, 1, -2) + b(1+2) + c (-2-2) = O Also the required plane is perpendicular to the plane 2x+y-z+6 = 0. :. their normals are perpendicular, then by perpendicularity condition 2a+b-c = O Solving (1) & (2) a-I, b=-12, c = -10 The required plane is (x-1) 12 (y+2) i.e, x-12y-10z-5 = O Example 4: 10 (z-2) = o Find the equation of the plane through the point (0,-1,-1) , (4, 5, 1) and (3, 9, 4) Solve: The required plane equation is 12
  13. x X 6 4 3 10 20] (y+l) [20-6] + lox (y+l) IOx-14y+22z+8 = o 5x-7y+11z+4 = O i.e., Example —5: 2 5 18] Find the equation of the plane which passes through the point (3,-1,4) and is perpendicular to the line joining the points (2,3,5) and (1, -2,3) Solve: The equation of the required plane which passes through the point (3, -2, 4) is given by Since the required plane is perpendicular to the line joining (2,3,5) and (1,-2,3) Hence the normal line to the plane is parallel to the line joining (2,3,5) (1 Their D.R. 'S are proportional D.R. 'S of the required plane are a, b, c D.R. 'S of the line joining the points (2,3,5) and (l, -2, 3) are (-1, -5,-2) The required plane is -I(x-3) 5(y+2) 2 (z-4) = O x-3+5y+10+2z-8=O x+5y+2z-1 = O Example 6: Find the equation of the plane which passes through the point (1,-2,1) and is perpendicular to each of the planes 3x+y+z-2 = 0 and x-2y+z+4=0 Solve: The equation of the required plane which passes through (1,-2, 1) is a(X-1) + b(y+2) + c (Z-1) = The required plane is perpendicular to each of the planes. 3x+y+z-2 = 0 x-2y+z+4 = 0 13 (1) (2) (3)
  14. . by perpendicularity condition a-2b+c = O Solving (4) and (5) a k 7 3(x-1) 20+2) :. the required plane is 3x-3-2y-4-7z+7 = 0 3x-2y-7=O Example — 7: :. The normal line to the required plane is perpendicular to the normal lines of the plane c - -18k (4) (5) Find the equation of the plane which passes through the points (6, 2, -4) and (3,-4, 1) and is parallel to the line joining (1 and (-1,2,4). Solve: The equation of the required plane which passes through (6, 2, -4) is a(x-6) + b(y-2) + c(z+4) = 0 If also passes through (3,-4, 1) a(3-6) + b(-4-2) + c( 1+4) = 0 (1) -3a-6b+5c = O Also, the required plane is parallel to the line joining the points (1, 0, 3) and (-1,2, 4) . Normal line to the required plane is perpendicular to the line joining (1, 0, 3) and 2, 4) D.R. 'S of the required plane are (a, b, c) D.R. 'S of the line joining (l, 0, 3) and (-1, 2, 4) are (-2, 2, 1) ...(2) . by perpendicularity condition 2a+2b+c = 0 Solving (1) and (2) b a 16 7 a= -16k k 18 -7k :. the required plane is -16k (x-6) - 7k (y-2) -18k ( z+4) = O 16x+7y+18z 38=0 Example 8: Find the angle between the planes 2x-y+z = 6 and x+y+2z = 7 Solve: 2 The D.C. 'S of the normal to the plane 2x-y+z-6 = 0 are Uö'Uö'Uö 1 The D.C. 'S of the normal to the plane x+y+2z 7 = 0 are The angle between the two planes is same as the angle between their normals, 6 2 .•.O = COS -1 1/2 = 600 14
  15. Example 9: The foot of the perpendicular from the origin to a plane is (2,-1,2). Find the equation of the plane. Solve: Let the required plane be ax+by+cz+d = 0 Since P(2, -1, 2) is the foot of the perpendicular from the origin 0(0,0,0) to the plane. The line OP is perpendicular to the plane. [ :. The normal line to the required plane and Let the required plane be ax+by+cz+d = 0 Since P(2, -1, 2) is the foot of the perpendicular from the origin 0(0,0,0) to the plane. The line OP is perpendicular to the plane]. . The normal line to the required plane and the line OP are parallel. Their D.R'S are proportional .. D.R's of the required plane are (a,b,c) D.R. 'S of the line OP are . by parallelism condition . The required plane is 2x-y+2z+d = 0 Also, (2, -1, 2) lies on it . 4+1+4+d= 0 . 2x-y+2z-9= 0 is the required plane. Example 10: Find the equation of the plane which bisects at right angle the join of (1 and (3, 1, 6). Solve: Since the required plane bisects the join of A(l, 3, -2) and B(3, 1, 6), it passes through the mid-point of AB, namely (2, 2, 2) Since the plane cuts AB at right — angles, AB is normal to the plane. D.R.'S of AB are . The equation of the required plane is 2(x-2) — 2 (y-2) + 8 (z-2) = 0 x-y 4- 8 = 0. Example 11 : Find the coordinates of the foot of the perpendicular from the origin to the plane 2x-3y+z Solve: Let p Y], ZD be the foot of the perpendicular from the origin, Since p(X/, z] ) lies on the plane, we have 2x1-3Y1+z/ D.R. 'S ofthe line OP are Y], D.R. 'S ofthe plane are (2, -3, 1) By parallelism condition k 2 = 2k,. Y/ = -3k, . (1) —4k+9k+k=7 (1) 15
  16. 2 3 .•.XFI , -1 2 . The foot of the perpendicular is (1, 3 1 Example — 12: Find the distance between the parallel planes 2x-3y+6z+12 = 0 and 6x-9y+18z-6= 0 Solve: Let Y], z]) be any point on the plane 2x-3y+6z +12 = 0 Distance between the parallel planes = length of the perpendicular from (x/ , Y], z]) on the plane 6x-9y+18z-6 = 0 16X1 9)'/ +180 61 62 +92 +182 13(2X1 3)'/ +60) 61 21 13(-12) 61 21 42 2 21 Example 13: Find the equation of the plane through the intersection of the planes 3x-y+2z-4 = 0 and x+y+z-2 = 0 and passing through the point (2, 2. 1) Solve: The required plane 3x-y+2z-4 + (x+y+z-2) = 0 Where X is to be determined. Since the point (2,2, 1) lies on the required plane 3(2) 2+2(1) -4+ +2+1-2) = o 2 3 . The required plane is 3x-y+2z 4 2/3 (x+y+z-2) = 0 7x 5y +4z 8 = 0 Example 14: Find the equation of the plane passing through the intersection of the planes x+3y z = 4 and 2x+2y+2z = 1 and perpendicular to the plane x+y-4z = 0 Solve: PI : x+3y-z 4=0 P2.• 2x+2y+2z 1 = O Equation of the required plane is PI + X P2 = 0 (x+3y-z-4) + (2x+2y+2z 1) = 0 Since it is perpendicular to x+y-4z = 0 By perpendicularity condition, (1+21) + (3+21) 4 (-1 + 21) = o . The equation of the required plane is 5x+7y + 3z 6 = 0. 16
  17. Example 15: Show that the plane 2x + 35y- 39z + 12 = 0 , 6x+6y-7z-8 = 0 and 12x-15y+16z-28 = 0 have a common line Solve: The given three planes have a common line only when one of the three planes passes through the line of intersection of the other two planes. . + 12) + X +6y ( +2) x + + 35) y + (-71 -39)z + (12 81) -O For some value of X, the plane 12x-15y+16z-28 = 0 and the plane (1) must represent the same plane . The corresponding coefficient must be proportional (1) 6 -k 2 -k 35 (7 X +39) 12 8X 12 15 16 From the first two ratios, we get X = 28 25 9 For this value of X , each of the above ratios = Hence the three planes have a common line. Exercise: 11 9 1. 2. 3. Find the equation of the plane (a) through the point (1, 3, 4) and parallel to the plane 2x+3y-4z+8 = 0 [Ans : 2x +3y-4z+5] (b) Through the point (1, -1, 2) and parallel to the xy plane. [Ans : z-2 = Find the equation of the plane through the point (-1,2,-3) and perpendicular to the line joining (-3, 1, 4) and (5, 4, 1) [Ans : 8x+2y-3z-5 = 0] Find the equation of the plane through the points. [Ans : 2x+6y-3z + 6 = 0] [Ans : x-3y-6z + 8 0] [Ans : 5x+2y-3z-17 = 0] (iii) ( (3,4, 2) 4.Find the equation of the plane i. Passing through the point (-1,1,1) and (1,-1,1) and perpendicular to the plane x+2y+2z-5 [Ans.• 2x+2y-3z+3 = 0] ii. Passing through the points (1,2,3) and (-4,1,-2) and perpendicular to the Plane 7x+2y-z +3=0 [Ans: 40y & +78=0] iii. Passing through the points (1,0,2) and (1,1,1) and parallel to z axis. [Ans: x-i = 0] 5.Find the equation of plane i. Passing through (3,1,-3) and (1,-1,1) and parallel to the line whose D.R. 's are (l, 1,-2). [Ans.• x+y+z=l] ii. Passing through (1,-2,3) and (-1,2,-1) and parallel to the line whose D.R. 's are (2,3,4) [Ans: 2x-z+1=0] 6.Find the equation of the plane i. Through the point (1,0,-2) and perpendicular to the planes 2x+y-z = 2 and x-y-z = 3. [Ans : 2x-y+3z+4 = 0] 17
  18. ii. Through the point (-1,1,4) and perpendicular to the planes 2x-y-z = 2 and x+y-3z+1 = 0. [Ans: 4x+5y+3z+11=0] 7.Prove that the following four point are coplanar. 8.Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 12x-4y-3z = 169. [Ans.• 9.Find the root of the perpendicular drawn from the point (1,0, 1) on the plane x+2y+3z=6 and also it image in the plane 82 10 94 13 [Ans.• 7'7'7 7'7'7 10.Find the equation of the plane through the intersection of the planes x+y+z and 2x+3y- z+4=0, parallel to i) y axis ii) The line joining the points (-1,2,-3) and (2,3,4). [ (i) x+4z=7 (ii) 20x+31y 13z + 46=0] 11.Find the angle between the following pair of planes (i) 2x-y+z=6 ; x+y+2z=7 (ii) 6x-3y-2z=7 ; x+y+2z = -9 [Ans: T/2] [Ans: Cos- (4/21) ] 12.Find the equation of the plane through the intersection of the planes 3x+4y+5z+ 18 = 0 and 2x+3y-5z+11 = 0 and passing through (1,2,-1) [Ans: x+y+10z+7 = 0] 13.Find the equation of the plane passing through the line of intersection of the planes x+2y+3z+2 = 0 and 2x+3y-z+3=0 and parallel to line whose direction ratios are [Ans.• 4x+5y-9z+5 = 0] 14. The plane 4x-y+5z-7 = 0 is rotated through a right angle about its line of intersection with the plane 2x-3y+3z-5 = 0. Find the equation of this plane in its new equation. [Ans : 5x+25y+z+7=O] 15.Find the equation of the plane passing through the line of intersection of the planes 2x- y+5z-3 = 0 and 4x+2y-z+7 = 0 and parallel to the z axis. [Ans.• 22x+9y+32=O] 16.Find the equation of the plane passing through the line of intersection of the plane 2x- 5y+z=3 and plane x+y + 4z=5 and i. Parallel to the plane x + 3y + 6z= 1 ii. Perpendicular to the plane 2x+y-z = 5 [ Ans: (i) x+3y+6z=7 (ii) y+Z=1] 17.Prove that the planes x+2y+3z=6, 3x+4y+5z=2 and 5x+4y+3z+18 = 0 have a common line of intersection. 18.Find the distance between the parallel planes (i) 2y+2z+19=O (ii) 2x+3y-4z+15=O,• 12x+3y-4z+2=o [Ans.• (i) 9 (ii) ] 19.Find whether the points (2,3,-5) and (3,4,7) lie on the same side of the plane x+2y-2z = 9. [Ans: No] 18
  19. 20.Show that the point (1,1,1) and (-3,0,1) lie on opposite sides of the plane 3x+4y- 1>+13 = o. The Straight Line: Different forms of the equation of a straight line in space are (1)Non- Symmetric form: we know that two planes in general intersect in a line. Hence a line in space can be represented by two linear equations. a IX + b 1 y -I-ClZ-l-d1=O and C12X + b2Y+C2Z+d2=O (2) Symmetric form: Equation of the straight line passing through he point (x,y,z) and having D.R' S (l,m,n) is given by m n (3)Two point form: Equation of the line joining the points and is given by The plane and the Straight line: Let us consider the plane ax+by+cz+d=0 and the straight line n (1) (2) a. The straight line is normal to the plane, if b. The straight line is parallel to the plane if al+bm+cn = 0 c. The angle between the line and the normal of the plane 900 - 0 and 0 is the angel at which the line (2) is inclined to the plane (1), thus cos (900-0) - Sin O = 2 Coplanar Straight lines: Let us consider the lines 11 12 The lines are coplanar if ml 2 12 Equation of the, plane in which the lines lie is given by Y2 Yl Z2 ml 12 19
  20. Skew Lines: Two straight lines which do not lie in the same plane are called skew lines or non planar.Skew lines are neither parallel nor intersecting. Such line has a common perpendicular. The length of the segment of this common perpendicular line is called the shortest distance (S.D) between them. The common perpendicular line itself is called the S.D line. Shortest distance between Two Skew lines (S.D) : Let us consider the lines 11 12 Shortest distance S.D 12 Equation of the S.D. between them is ml n ml 2 x—X2 12 2 121711 ) m n Where, l, m, n are D.C 'S of the line of shortest distance Examples 1: Find the D.C's of the straight line x=2 y=3z Solution: The given line can be written as z 1 1 1 2 3 .. the D.R. 's of the line are 1, I .. the D.C. 's of the line are Examples 2: Find the D.C. 's of the line 2x+1 4y 3 2z 3 3) 1 2 7 Also find a point on it. 3 Examples 3: 1 Find the equation of the line joining the points (3,0,2) and (1,-2,3). 20
  21. Solve: Equation of the line joining two points is 2 2 Examples :4 1 Find the equation of the line passing through (1,2,3) and parallel to the line joining ( 2,- 3,4) and Solve: Equation of the straight line through (1,2,3)with D.R'S is given by x ly 2 z 3 a b D.R 'S of the line joining the points (2,-3,4) and (3,2,1) are (1,5,-3) :. By parallelalism condition k 3 a=k, b=5k, -3k :. the required line is 1 5 Examples 5: 3 Find the equation of the straight line of intersection of the planes 2x-3y+3z=4, x+2y-z = -3 in the symmetric form. Solve: Let the symmetric form of the common line be n The D.R.S of the normal to the plane are (2,-3,3) : (1,2,-1) The common line lies in the two planes, hence it is perpendicular to both the normals, By perpendicularity condition 21-3m + = O I +2m - n = 0 Solving, 1=-3, 111=5, To find a point on the straight line, we assume = 0 . 2x1 and xl+2Y'/ 1 Solving = - /7 and YF - 1 10 7 . symmetric form of line is 3 5 z 7 21
  22. Examples 6: Show that the line 2 3 Solve : The given plane is 4x+4y-5z = 0 Parallel to — 2 3 4 is parallel to the plane 4x+4y-5z=0. 4 . normal to the plane (1) is perpendicular to the line D.R'S of normal to the plane (4,4,-5) D.R'S of line (2) are ( :. by perpendicular condition 5(4) =8+12-20 =0 Examples 7: Find the equation of the plane that contains the parallel lines ...(2) 1 ...(2) ...(3) ..(5) 2 3 and 1 2 3 Solve: The given line are 1 1 2 2 3 3 Any plane containing the line (1) is of the form a(x-l) + b ( y-2) + c (z-3) = 0 The lines (1) and (2) are perpendicular to the normal of the plane D.R. 'S of the line (1) are D.R. 'S of the line (2) are D.R. 'S of Normal line of the plane are (a,b,c) . by perpendicularity condition a+2b+3c = O Since the required plane also has the point (3,-2,-4) . a(+3-1) + b(-2-2) + c(-4-3) = 0 2a-4b-7c Solving 4 and 5 a 14 + 12 b a 2 13 b k 8 k . a = -2k , 13k, -8k -2k ( x-1) + 13 k z-3) = O 2x-13y+8z = 0 is the required plane. 22
  23. Examples 8: Find the equation of the plane which contains the line x is perpendicular to plane 2x+7y-3z=1 Solve: The equation of the plane containing the line x 2 is given by a(x-0) + b(y+3) + c (z+5) = 0 The plane (2) is perpendicular to the plane 2x+7y-3z= 1 2x+7y-3z= 1 . The normal lines are perpendicular DR'S of plane (2) are (a,b,c) D.R. 'S of normal line of the plane are (2,7,-3). :. By perpendicularity condition 2a+7b-3c = 0 The line (1) lies in the plane (2) :. Normal line of the plane is perpendicular to the line (1) D.R'S of the line (1) are :. By perpendicularity condition a+2b+3x=0 solving (3) and (4) a k 9 a = -9K, b=3K, . The required plane is 9k(x) + 3k(y+3) + k (z+5) i.e., 9x-3y-z+14=0 Examples 9: Find the value of k so that the lines 3 2k 2 2 3 2 and which 3 ...(1) ...(2) ..(3) and 2 3k Solve: 1 may be perpendicular to each other. 5 D.R. 'S of the given lines are (-3,2k,2) and (3k, 1,-5) :. the lines are perpendicular 3(3k) + 2k(1) + 2(-5) = 0 10 k= 7 Examples:10 Find the coordinates of the point of intersection of the line with the plane 2x+4y-z+1 = 0 Solve: 3 ...(1) 4 Given 2 3 4 2x+4y-z+1 = 0 23
  24. Any point on the given line is P(l +2r, 2-3r,-3+4r) If P lies on the plane (2), then 2(1+2r) +4 (2-9) (-3+40 +1 4r-12r _4r +2 + 8+3+1=0 7 6 10 3 Examples 11: Find the angle between the lines 3x+2y-z-5 = 0 = x+2y-2z-3 and = O = 7x+10y-8z Solve: The D.R'S a,b,c of the line 3x+2y+z-5= 0 = x+2y-2z-3 are given by the equations 3a+2b+c=O and a+2b-2c=O The symmetric form of the line 8x-4y-4z=0; 7x+10y-8z=0 is Hence D.R. 'S are (2, 1,3) :. The angle between the lines is given by COS O = 62 +72 +42 22 +12 +32 -1 0=COS 202 Examples 12: Find the shortest distance between the lines 2 Solve: 2 202 y 5 3 2 2 3 39 x and 1 3 5 12 SIDI 12 2 ml 4 2 3 3 3 2 24 4 1 5 2 12 ml) =169 + =507 39 507 12 ml) + (15-2)2 169+169 2 11)
  25. Examples 13: Find the shortest distance and the equation of the line of shortest distance of the lines 3 16 Solve: Given lines z-10 x-15 and 3 7 x 8 y +9 z —10 3 16 7 x-15 y 29 z Y 29 z 5 8 5 5 5 3 8 Any point P on the line AB is (3r1 +8 , -If] 9,7rl+ 10) Any point Q on the line CD is (3r2 + 15, 8r2 +29, -92 +5) Let PQ be the shortest distance which perpendicular to AB and CD D.R. 'S ofthe line AB are (3, -16,7) D.R. 'S of the line CD are D.R. 'S of the line PQ are (3 7, 8r2+ 16r1+ 38, -92 - 7 rl 5) PQ L AB 3(3r2 +7) 16 5) 77 h + - -311 .PQ L rCD 16r1+38) 5 ( -5 h -7 n 7 h + 11 -25 Solving (3) & (4) & Q (9, 13, 15) ...(2) ...(3) (1 5/ +(13 7/ +(15 3/ The equation of the line of the shortest distance is 2 3 6 Examples:14 Find the equation of the image of the line 3 in the plane 2x-y + z + 3 = 0 Solve: 5 5)=0 2 The image of the line joining the images of any two points on the line. It will be advantages if we choose one of the points as the point A of intersection of the line. 3 2 5 and the plane2x y +z+3 = 0 as the image of the point A in the plane is itself . since A is a point on the line (1) its coordinates maybe assumed as (3r+1, 5r+3,2r+4) Since A lies on the plane (2) 25 ...(2) ...(3)
  26. 2(3r+1) - (9+3) + 2r +4) +3 = O putting r = -2, we get A (-5,-7,0) Now we consider another point on the line (1) Let it be Let the image of P be Q (a,b,c). By definition of image, the mid-point M of PQ lies on the plane (2) Coordinates of M are As M lies on plane (2) 2 2 2 2 2 2 2 2a-b+C+9= O D.R. 'S of PQ are (a-I, b-3, c-4) By definition of image, PQ is normal to plane (2) a 119 3 c 4 2 k 1 ...(5) (say) ...(6) from (5) k+l, b=-k+3, c=k+4 . (4) gives 2(2k+1)- (-1
  27. 6. Find the angle between the lines Z — and 2 2 1 7. Find the value of k, if the lines and 1 k 5 2 [Ans.• 4 3 — are perpendicular. k Cos 9 [Ans: k=2] 8. Find the value of k, if the lines x 2 y Iz 3 and k 3 2 k 3 k [Ans.• 9.Find the equation of the plane, which contains the two parallel lines z Ix 2 y +3 1 4 1 5 4 5 [Ans : 11x-y-3z = 281 10. Find the equations of the line through (1,2,3) and parallel to the line x+y+2z 1 = 0 ; y-2=-z+3 ] 11. Show that the lines 2 3 and 3 4 5 find the equation of the plane in which they lie. are coplanar and 1 [Ans : 6x-5y-z = 01 12. Prove that the lines 1 and are coplanar. Find 4 7 2 3 8 their common point and also the equation of the plane containing them. [Ans: : -11x+6y+5z+ 67=0] 13. Show that the lines 2x+3y-4z = 0 = 3x-4y+z and 5x-y-3z+12 = O = x-7y+5z -6 are parallel. 14. Find the shortest distance and equation of the shortest distance between the following skew lines. x —10 y 9 z +2 1 3 2 x +1 y —12 z 5 2 4 1 [Ans.• 27 5Uö,x 8 11 3 z 2
  28. (ii) 1 7 (iii) 3 3 2 1 6 1 2 1 [Ans.• 1 4 [Ans.• 2 2 3 1 2 3 5 4 1 15.Find the shortest distance between the lines 2 and 2x+5y-8z-52 = O = 3x 3y+2z+27 Sphere and Circle: [Ans.• 6$ ] A sphere is the locus of a point, which moves in space such that its distance from a fixed point is constant. The equation of the sphere whose centre is (a, b, c) and radius is r, is given by (x-a)2 + (y-b2) + (z-c)2 2 The standard form of equation to a sphere is give by x +y +z 2ux +2vy+2wz + d = 0 with centre (-u, -v, -w) and radius r = The equation of the sphere joining the end point z]) and (x2, Y2, z2) of its diameter is given by (x-xD (x-X2) + (y-yD (y-Y2) + ( z-zD (z-Z2) = 0 Tangent line and Tangent Plane: When a straight line intersects a sphere at two coincident points or when it touches a sphere at a point P, the line is called a tangent line of the sphere at P and is perpendicular to the radius of the sphere through P. There are many such tangent lines lie on plane through P, which is perpendicular to the radius of the sphere through P. This plane is called the tangent plane of the sphere at p. Equation of the tangent plane to the sphere x +y +z + 2ux +2vy+2wz + d = 0 at P Y], z] ) is xxl+yy'l+zz/+u (x+X1) + v ( y+Y1) + w (z+Z1) + d = 0 Plane Section of a sphere: The points common to a sphere of radius r and a plane at a distance h from its centre lie on a circle of radius r2 —h 2 with its centre at the foot of the perpendicular from centre of the sphere to the plane. This circle is a great circle, if the plane passes through the centre of the sphere. Note: The intersection of two sphere is also a circle. The circle of intersection in this case, is jointly represented by the spheres Sl and S2 as Sl — S2 = 0 28
  29. Equation of a sphere through a circle: Let the given circle be represented jointly by the equations S: x + y + z 2ux + 2vy + 2wz + d = 0 and P: ax + by + cz +d = 0 as S+IP=O Also, Sl + AS2 = 0, Sl & S2 are spheres. Orthogonal Spheres: Two orthogonal spheres are such that the tangent planes to the spheres at any common point are at right angles. If the two spheres X -I-y + Z + 2111 X +2 VI y +2 WI Z d 1 = O and x +y +z + 2112 x +2v2 y+2W2 z + 612 = 0 are orthogonal, then Note:l. 1. Two spheres touch each other if there is a common point of intersection of the spheres at which the tangent planes of the spheres coincide. 2. Two spheres touch one another externally, if the distance between their centres is the same as the sum of their radii. 3. Two spheres touch one another internally, if the distance between their centres is the same as the difference of their radii. Examples 1: Find the centre and radius of the sphere 2x2+2y2+2z2-6x+8y-8z 1 = 0. Solve: Given sphere is 1 2 = 42 3 Centre i.e., radius = 2 3 2 4, 2w— 2 3 2 9 1 4 4 9+16+16+1 4 Examples 2: 1 2 2 1 2 1 42 2 Find the equation of the sphere whose centre is (1,2,3) and radius 4. Solve: (x-1)2 + (y-2)2 + (z-3)2 i.e., x +y +z2-2x-4y-6z-2= 0 29
  30. Examples 3: Find the equation of the sphere having the centre (3,-2,2) and passing through the = 45 point (-1,3,4). Solve : (Radius)2 2 [Distance from (3,-2,2) to (3+1)2 + (-2-3)2 + (2-4)2 45 :. Equation of the sphere is (x-3)2 + (y +2) 2 + (z-2)2 . •.x2 +y2 +z2-6x+4y-4z-28 Examples 4: Find the equation of the sphere which has the line joining the points (2,7,5) and (8,-5,1) as diameter Solve: The equation of the sphere is (x-2) (x-8) + (y-7) (y+5) + (z-5) (z-1) = 0 i.e., x2+y2+z2-10x-2y-6z-14 = 0. Examples 5: Find the equation of the sphere whose centre is (6,-1,2) and which touches the plane 2x-y+2z=2 Solve: Radius of the sphere is the perpendicular distance of (6, -1,2) from the plane 2(6) (-1 2 Radius = + 22 + 12 +22 =5 . The equation of the sphere is (x-6)2 + (y+1)2 + (z-2)2 = 52 x2+y2+z2-12x+2y-4z+16 = 0 Examples 6: Find the equation of the sphere through the four point (0,0,0) (a,0,0) , (0,b,0), (0,0,c). Solve: Let the equation of the sphere be x +y +z +2ux+2vy+2wz+d — 0 Since (1) passes through (0,0,0) Similarly, it passes through (a,0,0), (0,b,0), (0,0,c) a +2ua = 0 b2 + 2vb = 0 c +2wc=0 i.e., u — a 2 b 2 . The sphere is x +y +z -ax-by-cz = 0 Examples 7: Find the equation of the sphere which passes through the points (3,4,2), (2,0,5), 30
  31. Solve: Let the equation of the sphere be x +y +z +2ux+2vy+2wz+d Since (3,4,2) lies on the sphere 32+42+22+2(U) (3) + 2 v (4) + 2(w) (2) + d = O Similarly it passes through (2,0,5) , (2,4,5), (3,3,1) 4u+8v+10w+d = -29 6u+6v+2w+d = -19 . u= -1, v=-2,w=-3, d=5 Thus the sphere is x2+y2+z2-2x-4y-6z+5 = 0 Examples 8: 0 Find the equation of the sphere passing through the points (1,1,-2), (-1,1,2) and having the centre of the sphere on the line x+y-z-l = 2z-y+z-2. Solve: Let the equation of the sphere be x +y +z +2ux+2vy+2wz+d — 0 It passes through the points (1,1,-2) and (-1,1,2) :.2u+2v-4w+d = - 6 -2Ll+2V+4VV-l-d = -6 . The center of the sphere (1) lies on the line determined by the two planes x+y-z-l = 0 and 2x-y+z-2 = 0 (2) ...(3) Centre of the sphere (1) is (-u,-v, -w) -2u +v w = 2 (2) — (3) given, 4u-8w = 0 i.e., u = 2w :. (4) and (5) -w-v = 1 -5w+v = 2 from (4) we get u = -1 from (2) we get d = -5 ...(5) . The equation of the sphere is x +y +z -2x-y-z-5 = 0 Example 9: Show that the plane 2x-2y+z+12=0 touches the sphere x +y +z2-2x —4y+2z and Find also the point of contact. Solve: Given that x +y2+z2-2x-4y+2z = 0 x2 2y+z+12 = 0 Centre of the sphere (1) is (1,2,-1) RadiLlS is I 4 -k I + 3 3 Length of perpendicular from (1,2,-1) on plane (2) 4+4+1 =3 31 ...(2)
  32. i.e., Length of perpendicular from (1,2,-1) to the plane (2) = radius of the sphere . The plane (2) touches the sphere (1). Let P be the point of contact. Then CP is normal to the plane (2) i.e., Normal line of the plane (2) and the line CP are parallel. :. Their D.R. 'S are proportional . D.R.'S of CP are Also CP passes through C(1,2,-1) Hence equation of CP is 2 2 1 any point on the line CP is (2r+1,-2r+2, r-l) if this point lies on the plane (2), it will be the point of contact. Then 2(2r+1) 2 (-2r+2) +(r-l) +12 = O :.P the point of contact is (-1,4,-2) Examples 10: touches the sphere x +y +z2-2x-4y-4=0. Show that the line 3 4 5 Also find the coordinates of the point of contact. Solve: Given 3 4 5 ...U) ...(2) Any point on the line (1) is (3r+6, 4r+7, 5r+3) This point lies on the sphere (2) (3r+6)2 + (4r +7)2 + (5r+3)2 2 (3r+6) _4 (4r+7) 4 = i.e., r -I-2r-l- 1 = O Thus the given line and the sphere have exactly one common point, namely (3,3,-2) is the point of contact. Examples 11: Find the equation of tangent planes of the spheres x +y +z -4x-4y-4z+10 = 0 which are parallel to the plane x z= 0 Solve: Let be the point on the sphere at which the tangent plane is drawn. The equation of the tangent plane at Y], ZD is xxl+yy'l+zzl - 2(x+xD 2(y+yD 2 (z+Z1) + 10=0 i.e., (xl 2) x + (Yl 2) y+ (Zl 2) = -2x1-2Y1 20 + 10=0 This plane is parallel tox z = 0 (1) 1 k (say) 1 . x/ = k+l, Y'/ = 2, Since lies on the sphere, we have (1
  33. from (2), the points are (3,2,1) and (1,2,3) . form (1), the equation of the tangent planes are x-z-2 = 0 and x +z Examples 12: Find the centre and radius of the circle given by x + y + z 2x-4y-6z and x+2y+2z-20 = O Solution: Given, x +y2+z2-2x-4y-6z-2=0 x+2y+2z-20=O Centre and Radius of the sphere (1) is C (1,2,3) and Radius 1+4 +9 +2 ...(1) ...(2) C2, the centre of the given circle is the foot of the perpendicular from Cl on the given plane (2) . Cl C2 is normal to the plane (2) and parallel to the normal line of the plane (2) D.R.'S of Cl are . Equation of Cl C2 are x ly 2 z 3 1 2 2 Any point on the line CIC2 is (r+l, 2r+z,2r+3) If the co-ordinates in (3) represent C2, they should satisfy the plane (2) . C2 lies on the plane (2) . (r +1) +2 (2r +2) + 2(2r +3) . length of CIC2= (2 =3 if r is the radius of the circle = 20 2 = 16-9 . centre and radius of circle are (2,4,5) and d'. Examples 13: Find the equation of the sphere which pass through the circle x +y +z -2x+2y+4z-3 = 0; 2x+y+z-4 = 0 and touch the plane 3x+4y-14 = 0. Solve: Given circle is the intersection of the sphere and the plane S:x2+y2+z2-2x+2y+4z-3 = 0 P:2x+y+z-4 = 0 Then S+XP = 0 represents a sphere passing through the circle determined by (1) & (2) ...(3) (1) ...(2) i.e., 33 (3)
  34. Centre and radius of the sphere (2) are (2+1) ( —4 + X) 2 2 2 2 radius 2 +(3+4X) 2 Since the sphere touches the plane 3x+4y-14 = 0 the perpendicular distance from the centre of the sphere to this plane is equal to the radius of the sphere 2 2 x 14 i.e., 32 +42 212 - 0 2 +(3+4X) 2 using these values in (3) --11=0 Examples 14: Find the equation of the sphere passing through the circle x +y +z +2x-4y+2z=3; x+2y+3z = 6 and through the point (2,0,1) Solve: Given x +y +z +2x 4y+2z 3=0 x+2y+3z 6 = 0 Equation of the sphere through the circle (1) is 5 + XP = 0 Since the sphere passes through (2,0, 1) X ( 2+0+3-6) . (2) gives X2+y2+Z2 + 2x + 2z 3 + 8 (x+2y+3z-6) X + + + -1-12y+26Z 51 = is the required sphere. Examples 15: Find the equation of the sphere having the circle x + y +z + 10y 4z 8 = a great circle. Solve: The equation of any sphere passing the give circle is of the form , X-I-y+Z=3 as If the given circle is a great circle of sphere (1) centre of sphere (1) should be on the plane x+y+z-3= 0 in which the given circle lies. . Centre of sphere (1) is 2 2 1 2 . Equation of the required sphere is x 2 3 34 8z +4=0
  35. Examples 16: Find the equation of the sphere through the circle x -By +z2 + 10y 4z 8=0 as a great circle Solve: Given Sl — S2 = 0 represent a plane HenceSl=O S - 2 — 0 determines a circle . Sl + X (Sl S2) = 0 represent the equation of the sphere passing through the circle determines by Sl = 0 and Sl — S2 = 0 (1) 11 , -8 its centre is 2 2 2 Since the cross section is a great circle this centres lies on the plane x+y+z 3 2 2 2 From (1), Required sphere is x + y + z2—4x+ 6y 8z+ 4 = 0 Examples 17: Prove that the two sphere x + y + z +4y 4z = 0 and x + y + + 10x + 2z + 10 = 0 touch each other and find the point of contact. Solve : Given (1) ...(2) Centre and radius of sphere (1) is Cl (1,-2,2) and rl = 3 Centre and radius of sphere (2) is C2 (-5,0,-1) and r2 = 4 7 We see that = rl + r2 :. The two sphere touch each other externally. Also the point of contact P divides CIC2 internally in the ratio rl : r2 i.e., 3:4 -15+4, 0-8 , Hence P 3+4 3+4 33++84) i.e., Pis 7 Examples 18: 7 Prove that the two sphere x + y + z + 6y + 2z + 8 = 0 Solve: From the equation of spheres — 0 intersect each other orthogonally. , 2 = 20 35
  36. = o +24+4- (8+20) Hence the two spheres intersect orthogonally. Exercise: 1. Find the equation of the sphere whose (i) Centre is (2,-1,0) ; radius is 4 (ii) Centre is (1 ; radius is 3 [Ans.• (i) x2 + Y2 + —4x+2y 11 = 0 (ii) x2 + y 2 + z2 2X+8y+2Z+12=0] 2. Find the radius of the sphere whose centre is (4,4,-2) and which passes through origin. [ Ans.• r = 6] 3. Find the Centre and radius of sphere (iii) (iv) (v) (vi) x + y + z2-2x-3y+4z+5 = 0 x + y 2 + z2-4z+6y-8z+4 = 0 2x2 + 2y2 + 2z2 +5x+7y+8z-1 = O 16(x2 + Y2 +z2) 16x-8y-16z 55=0 [Ans: (i) (1,3 , 2); 3 2 (iii) (—5 742), 5 1 11 (iv) 2'4'2 4 4. Find the equation of the sphere through the four points (vii) (viii) (0, O, 0) x + y 2 + z2 3x-3y-3z+6 = 0 7x2 +7 + 7z 2 15x 25y 11z=01 (ii) 5. Find the equation of the sphere which passes through the points (1, 0, 0 ), (0, 1, 0), (0, 0 , 1) and has its centre on the plane x+y+z=6 6. Show that the plane 2x-2y+z+16 = 0 touches the sphere x + y + z2 + 2x-4y+2z = 3 and find the point of contact. [Ans: (-3, 4, -2) ] 7. Find the equation of the circle which lies on the sphere x + y + z2-6x-8y+2z 9= 0 and whose centre is (1, 2, -1) [Ans.• x2 + y 2 + z2 6x 8y+2z-9 = 0 ; x+y-3 = 0 ] 8. Find the equation of the sphere having its centre on the line 5z + 2x=0 = 2y 3z and passing through the points (-4,0,-2) and (-1,2,-1). [Ans.' x2 + y 2 + z2+10x-6y 4z + 12 = 0] 9. Find the equation of the sphere which passes through the circle x2+y2+z2-4=0; 2x-y+2z 3 = 0 and the point (2,1,1). [ Ans.•x + Y2 + z2-2x+y-2z=1] 10. Find the equation of the sphere which passes through the point (1, -4,3), (1, -5,2) and has its centre on the line 4 1 3 [Ans.' x + + z2 _4x+7y 3z + 15 = 0] 36
  37. 11. Find the equation of the sphere (i) Which has its centre on the plane 5x+y-4z+3 = 0 and passing through the circle x2 + y + z2-3x-4y-2z+8 = 0; 4x-5y+3z-3 = 0. (ii) Through the circle x + y + z2 + 2x+3y+6 = 0 ; x-2y +4z-9 = 0 and centre of the sphere x + y + z 2x+4y-6z+5 = 0 x + Y2 + z2 + 7y-8z+24 = 0 (ii) x2 + + + 7y 8z + 24 = 01 12. Find the centre and radius of the circle given by x + y + z +2x-2y-4z-19 = 0 and x+2y+2z+7 = 0 75 2 3'3'3 13. Find the centre and radius of the circle given by a. x + Y2 + z +7 y 2z+2=0 ;2x+3y+4z = 8 c. x + y + z 2 2x-4z-11 = 0; 2z+y+z= 15 (i) (0, 72 J); 2 Ans.• (iii) (1, 0, 2); d' 14. Find the equation of the sphere having the circle x + y + z +6x+3y-z=8 ; x+2y-2z+6 = 0 on it as a great circle [Ans: 9(x2 + + z2) + 52x +23y-5z 84 = 15. Show that the circles whose equations are x + y + z -3x-4y+5z-6 = 0 ; 80 x+2y-7z= Ox + y + z2-2x+3y+4z-5=0 ; 5y+6z+1 = 0 lie on the same sphere and find its equation [ Ans.• x2 + Y2 + z2-2x-2y-2z-6 = 0 ] 16. Find the equation of the sphere which pass through the circle x + y + z2-2x-4y = 0, x+2y+3z=8 and touch the plane 4x+3y=25 [ Ans.• x2 + Y2 + z2 + 6z = 16] 17. Show that the plane 2x +2y+z+4 = 0 touches the sphere x + y + z 2x—4y +2z = 3 and find the point of contact . [Ans: ] 18. Find the equation of the sphere whose radius is 2 and which passes through the circle of intersection of the spheres LSI ; x + y + z + 2x+2y-2z-6 [Ans.• 10 = 011 3 19. Find the equation of the sphere through the circle x + y + z +2x+3y+5z = 0 ; 2x+6y+5z-6 = 0 and passing through the centre of the sphere [Ans.• x + y + z2 + 4x + 9y + 10z 20. Find the equation of the sphere that passes through the circle x + Y2 + z2-2x+3y-4z+6 = 0 ; 3x-4y +5z-1 = 0 and cuts the sphere x + y + z + 2x+4y-6z+11 = 0 orthogonally. [Ans: 5(x2 + Y2 +z2) 13x +19y 2±+45 = 0] 37
  38. 21. Find the equation of the sphere which touches the plane 3x+2y-z+2 = 0 at the point (1, -2, 1) and cuts orthogonally the sphere x 2 + y 2 + z2-4x+6y+4 = 0 [Ans.•x + y 2 + z2 + 2x 2y +4z-3 = 0 22.Find the equation of the sphere that passes through the circle x + y + z +x-3y+2z 1 = 0 2x+5y-z+7= 0 and cuts orthogonally the sphere [Ans.• 2(x2 + Y2 + z2) + 8x +9y+z+19 = 0]. 38

Need a Tutor or Coaching Class?

Post an enquiry and get instant responses from qualified and experienced tutors.

Post Requirement

Related Study Notes

Upload Study Notes

If you have your own Study Notes which you think can benefit others, please upload on LearnPick. For each approved Study Note you will get 100 Credit Points and 100 Activity Score which will increase your profile visibility.

Upload Now
Home > Study Notes > Analytical Geometry