This booklet consists of basic definitions, different forms of straight lines, formulas and practice questions on the discussed topics.
1
Compiled by : Dipu Sir STRAIGHT LINES BASIC CONCEPTS 1. 2. 3. 4. The Straight Liné : The every equation of first degree in x and y represents a straight line. Slope of a Line : The slope of a line is the tangent of the angle, measured anticlockwise from the positive direction of x-asix, to the line. If the inclination of the line is 0, then slope of the line, (m) tan 0. Different Forms of Staight Line : O (i) Euation of axes : The equation of x-axis is y = 0 The equatioq of y-axis is x = 0. 0 o x x (ii) Equation of straight lines parallel to axes : The equation of the straight line parallel to x-axis and at a distance of b units from it is y = b The equation of a straight line parallel to y-axis and at a distance of 'a' units it is, x = a. (iii) : The equation of a straight line which makes an angle 0 with the x-axis and which has intercept c on the y-axis is y = mx + c, where m = tan 6. (iv) Intercept from : The equation of a straight line making intercepts a and b on x-axis and y-axis respectively, is 1+1=1 (v) Normal from : The equation of a straight line which has a perpendicular makes an angle a with the positive direction of x-axis is x cos a + y sin a = p (vi) One-point form : The equation of a straight line passing through Yl) having gradient mis y—Yl = (vii) Two-point from : The equation of a straight line passing through the points (XP Yl) and (x? Y2) is Y-YI = x (viii) Distance from : The equation of a straight line passing through Yl) and making and angle 9 with x-axis 1 is =r where r is the distance of any point on the line from the point (XP y,) cos0 sino Hence the coordinates of any point on the line are given by x = x I + r cos e and y = yr + r sin 0. Angle between two lines : Let the equations of two straight lines be y m x + c , and ml = tan Or, rn2 = tan Or Let 9 be the angle between the two straight lines. 01 = 0+02 —O 2 tan 0 = tan (91 — 02) tano — tano 2 or tan 0 = 1+ tano tano 1+mm 2 12 If lines are parallel, then 0 = 0 2 = 0 ml — rn2 = 0 ml = m2 I + mm 12 2 o 1 x
2
If lines are parallel, then 6 = 2 2 I + m mi mime = —1. 1+mm 12 5. Condition for the concurrency of three straight lines : 1st Mathod : Find the point of intersection of any two lines and show that it satisfies the third also. 2nd Method : If P = 0, Q = 0, and R = 0 equations of the given lines and if IP + mQ + nR = 0 takes the form ox + oy + oz = 0 where l, m and n are any three constants to be found by inspection then the three given lines are concurrent. 3rd Method : The three straight lines atx + bjy + ct = 0, a2X + b2Y + = O and + b3Y + = O a 1 are concurrent if a 2 b 1 2 b 3 1 3 6. 7. 8 Perpendicular distance of a point from a line : The Peropendicular distance of a point (XP Yl) from the line ax + by +c ax + by + c = 0 is given by Bisector of angles : The equations of the straight lines bisecting the angles between the lines ax + b y +c 1 all + b y + = 0 and + b2Y + = 0 are 2 1 Note : If in (i), cr and cr are of same sign, then (a). The angle between the given lines in which origin lies is acute, if ala2 + blb2 > 0 (b) The positive sign of R.HS. of (i) will give the bisector of Acute angle, if ala 2 + bLb2 < 0 obtuse angle, if ala 2 + bib2 > 0 2 2 2 Image (x, y) of point (XP y,) about the line ax + by + c = 0 is given by 2 (ax + by +c) a b and foot (a, B) f point (x p Yl) on the line ax + by + c = 0 is given by 13 —Y (ax + by +c) (i) ax + by + c a b
3
9. Equation of line making angle a from a line of slope tan O and passing through (h, Yl) is y axis B(x, y) A(xt, YD x axis y —Yl = tan (0 ± a) (x — Xi) Locus Locus is set of points, which satisfy given condition (s). General method to find the læus : (i) Take the co-ordinates of P as (h, k), (a, ß) or (x I, Yl) (ii) Unisng the equations given / the conditions given, find an equation in h and k by eliminating the various paramenters given. Generalise (h, k) by replacing h by x and k by y. The equation thus obtained is the locus of the required point p. Position of a point (xt, Yl) with respect to a line ax + by + c = 0 The point (xj, Yl) lies on the same side or opposite side of a line ax + by + c = 0 ax I + by I +c > 0 or
4
ml + and b a b Point of Intersection of Lines Let S ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 represents a pair of straght lines, then differentiate it with respect to x (assuming y constant) = 2ax + 2hy + 2g = 0 Now differentiate it with respect to y•(assuming x constant) — = 2by + 2hx + 2f = O (i) (ii) Solution of equation (i) and (ii) gives the intersection point of lines. Angle between Pair of straight Lines The angle 0 between the lines represented by S = ax2 + 2hxy+ by2 = 0 and is given by tan e = a+b (i) If a + b = 0, then lines are perpendicular. (ii) If 112 = ab, then lines are parallel. Slope of Lines S ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 Consider only two degree homogeneous part of this equation, i.e., ax2 + 2hxy + by2 = 0 2 or b x (on dividing it by x2) x (Here — represent slopes of lines) x Equation of Angle Bisectors Equation of angle bisectors of the angles between the lines represented by the equation x2—y2 _ xy ax2 + 2hxy + by2 = 0 is a—b h Point of Intersection of the Lines The point of intersection of bg—hf af—gh ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is h2-ab' h2-ab Homogenization Let curve is ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 Now line Ix + my = I intersect it at two points A and B, then equation of pair of straight lines passing through origin and intersection points of curve and line AB i.e. OA and 0B is ax2 + by2 + 2hxy + 2gx (Ix + my) + 2fy (Ix + my) + c (Ix + my)2 y axis o x axis
5
Angle between the lines which do not pass through origin : The angle between the lines represented by the equation ax2 +by2 + 2hxy + 2gx + 2fy + c = 0 is given by -1 e = tan a+b The lines are perpendicular, if a + b = 0 The lines are parallel, if hi = ab Translation of Axes Transformation of axis is done to reduce calculation. It is of two types : (i) Tranilation of Axis : In this, origin is shifted and that changing the direction of axes by using the substitution . (ii) Rotation of Axes : In this, axes are rotated through some angle, say, 0 by using the substitutions x = X cos 0 -- Y sin 0 y = X sin 0 + Y cos 6 Important Points Area of triangle whose sides are arx +bry + cr = 0, r = l, 2, 3 -2 0 0 x x x x where q, C and C3 ae cofactors of q, and respectively in the determinant is 21CCCl 123 2 a 3 b 1 b 2 b 3 a 1 c 1 •c 2 b 1 b 2 b 3 2 c 1 c 2 = I(alb2 — a2bl) (a2b3 a3b2) (agbl — atb3)l A point (x] , h) lies between two parillel lines ax + by + = 0 and ax + by + c2 = 0 if (ax I + by I + q) (ax I + byl + % ) < () Equation of line parallel to lines ax + by + = 0 and ax + by + = 0 and equidistant from the lines is 2 = 0 as m: n = 1 : I ax + by + 2
Discussion
Copyright Infringement: All the contents displayed here are being uploaded by our members. If an user uploaded your copyrighted material to LearnPick without your permission, please submit a Takedown Request for removal.
Need a Tutor or Coaching Class?
Post an enquiry and get instant responses from qualified and experienced tutors.
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 25 Credit Points and 25 Activity Score which will increase your profile visibility.