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Engineering Mathematics

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Published in: Mathematics
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Definitions and Solutions of differential equations!

Hari B / Hyderabad

9 years of teaching experience

Qualification: M.Sc (jntuh campus - 2012)

Teaches: Algebra, Business Mathematics, Mathematics, Statistics, B.Tech Tuition, Polytechnic

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  1. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. PARABOLA The locus of a point which moves in a plane so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line is called a conic section or conic. The fixed point is called focus, the fixed straight line is called directrix and the constant ratio 'e' is called eccentricity of the conic. If e = 1, then the conic is called a parabola. If e < 1, then the conic is called an ellipse. If e > 1, then the conic is called a hyperbola. The equation of a conic is of the form ax + 2hxy +by + 2gx + 2fy + c = 0. A line L = 0 passing through the focus of a conic is said to be the principal axis of the conic if it is perpendicular to the directrix of the conic. The points of intersection of a conic and its principal axis are called vertices of the conic. If a conic has only one vertex then its centre coincides with the vertex. A conic has at most two vertices. The midpoint of the line segment joining the vertices of a conic is called centre of the conic. A conic is said to be in the standard form if the principal axis of the conic is x-axis and the centre of the conic is the origin. The equation of a parabola in the standard form is y = 4ax. For the parabola y = 4ax, vertex=(0, 0), focus=(a, 0) and the equation of the directrix is x+a=0. If we rotate the axes 900 in the clockwise direction then the equation y = 4ax of a parabola is transformed to x = 4ay. For the parabola x = 4ay, vertex = (0, 0), focus = (0, a), the equation of the directrix is y+a = 0 and the equation of the principal axis is x = 0 (y-axis). A point (Xl, YD is said to be an i) external point of the parabola y = 4ax if — 4aX1 ii) internal point of the parabola y = 4ax if — 4aX1 < 0. A chord passing through a point P on the parabola and perpendicular to the principal axis of the parabola is called the double ordinate of the P. A chord of the parabola passing through the focus is called a focal chord. A focal chord of a parabola perpendicular to the principal axis of the parabola is called latus rectum. If the latus rectum meets the parabola in L and L' , then LL' is called length of the latus rectum.
  2. Parabola 20. The length of the latus rectum of the parabola y 2 = 4ax is 41al. 21. If P is a point on the parabola with focus S, then SP is called focal distance of P. 22. The focal distance of P(XI, YD on the parabola y 2 = 4ax is + al. 2 23. The equation of the parabola whose axis is parallel to x-axis and vertex at (u, ß) is (y - ß) 4a(x -u), (00). 2 24. For the parabola (y — ß) = ± 4a(x — u), the focus is (u±a,ß) and the equation to the directrix is 25. The equation (y — = ±4a(x — u) can be put in the form x = ly + my+ n. 26. The equation of the parabola whose axis is parallel to y-axis and vertex at (u, ß) is (x — 002 = ± 4a(y— ß) 27. For the parabola (x - u) 2 = ± 4a(y — ß), the focus is (u, ß ±a), the equation of the directrix is y = 28. The equation (x -u) 2 = 4a(y - ß) can be put in the form y = Ix + mx + n. 29. We use the following notation in this chapter S y 2 — 4ax, Sl yyl — 2a(x+x1), Sli = S(XI, YD —4ax1, Sn YlY2 30. Let P(XI, YD be a point and S y 2 — 4ax = 0 be a parabola. Then i) P lies on the parabola Sli = 0 ii) P Lies inside the parabola Sli < 0 iii) P lies outside the parabola Sli > 0 — 2a(X1 + X2). 31. The equation of the chord joining the two points A(XI, YD, B(X2, Y2) on the parabola S = 0 is Sl+ S2 = S12. 32. Let S= 0 be a parabola and P be a point on the parabola. Let Q be any other point on the parabola. If the secant line PQ approaches to the same limiting position as Q moves along the curve and approaches to P form either side, then the limiting position is called a tangent line or tangent to the parabola at P. The point P is called point of contact of the tangent to the parabola. 33. If L = 0 is a tangent to the parabola S = 0 at P, then we say that the line L = 0 touches the parabola S= 0 at P. 34. The equation of the tangent to the parabola S = 0 at P(XI, YD is SF 0. 35. Let S = 0 be a parabola and P be a point on the parabola S = 0. The line passing through P and perpendicular to the tangent of S = 0 at P is called the normal to the parabola S = 0 at P. 36. The equation of the normal to the parabola y 2 = 4ax at P(XI, YD is Yl(x — Xl) +2a(y — YD = 0. 37. The condition that the line y = mx + c may be a tangent to the parabola y = 4ax is c = a/m.
  3. Parabola 38. The equation of a tangent to the parabola y = 4ax may be taken as y = mx + a/m. The point of contact is (a/m2 , 2a/m). 39. If ml, nu are the slopes of the tangents of the parabola y = 4ax through an external point P (Xl, YD, then ml + nu = YI/XI, minu = a/X1. 40. The line joining the points of contact of the tangent to a parabola S = 0 drawn from an external point P is called chord of contact of P with respect to the parabola S = 0. 41. The equation to the chord of contact of P(XI, YD with respect to the parabola S = 0 is Sl = 0. 42. The locus of the point of intersection of the tangents to the parabola S = 0 brawn at the extremities of the chord passing through a point P is a straight line L = 0, called the polar of P with respect to the parabola S =0. The point P is called the pole of the line L = 0 with respect to the parabola S = 0. 43. The equation of the polar of the point P(XI, YD with respect to the parabola S = 0 is SF 0. 44. If P is an external point of the parabola S = 0, then the polar of P meets the parabola in two points and the polar becomes the chord of contact of P. 45. If P lies on the parabola S = 0, then the polar of P becomes the tangent at P to the parabola S=O. 46. If P is an internal point of the parabola S = 0, then the polar of P does not meet the parabola. 47. The pole of the line Ix + my + n = 0 (l with respect to the parabola y 2 = 4ax is (n/ l,—2am/ l). 48. Two points P and Q are said to be conjugate points with respect to the parabola S if the polar of P with respect to S = 0 passes through Q. 49. The condition for the points P(XI, YD, Q(X2, Y2) to be conjugate with respect to the parabola S = 0 is 0. 50. Two lines Ll = 0 , 122 = 0 are said to be conjugate lines with respect to the parabola S = 0 if the pole of Ll = 0 lie on 51. The condition for the lines IIX + nuy + m and 1 2x + my + n2 = 0 to be conjugate with 52. The equation of the chord of the parabola S = 0 having P(XI, YD as its midpoint is Sl = Sil. 53. The equation to the pair of tangents to the parabola S from P(XI, YD is Sl = SMS. 54. A point (x, y) on the parabola y = 4ax can be represented as x=at , y =2at in a single parameter 2 t. These equations are called parametric equations of the parabola y = 4ax. The point (at2, 2at) is simply denoted by t. 55. The equation of the chord joining the points ti and t2 on the parabola y = 4ax is y(tl + t2) = 2x + 2at1t2• 56. If the chord joining the points ti and t2 on the parabola y = 4ax is a focal chord then t1t2 — 2 57. The equation of the tangent to the parabola y = 4ax at the point 't' is yt = x + at .
  4. Parabola 58. The point of intersection of the tangents to the parabola y = 4ax at the points ti and t2 is (atlt2, a[t1+t2]). 3 59. The equation of the normal to the parabola y = 4ax at the point t is y + Xt = 2at + at . 60. Three normals can be drawn from a point (Xl, Yl) to the parabola y = 4ax. 61. If ti, t2, t3 are the feet of the three normals drawn from point (Xl, YD to the parabola y 2a - Yl then ti + t2 + t3 = 0, tit2 + t2t3 +t3t1 = '1 23 a a 62. If the normals at ti and t2 to the parabola meet on the parabola, then t1t2 = 2. = 4ax 63. For the parabola x = ly + my + n, 2 2 I-m m -m Vertex= n Focus = 41 ' 21 41 1 -m 21 2 I-m axis is y+ E directrix is x=n+ Latusrectum = 21 64. For the parabola y= Ix + mx + n, Vertex= 2 m m ,Focus = —,n 21 41 m I-m 21 1 41 2 41 2 I-m Latusrectum = directrix is y=n+ axis is y + 41 65. The condition that the line Ix +my + n = 0 may be a tangent to the parabola i) y =4ax is am = In ii)x =4ay is al 2 =mn. 66. The pole of the line Ix+my + n = 0 with respect to the parabola n i) y 2 - 4ax is 2am 2al n ii) x = 4ay is 67. The length of the chord joining tl, t2 on y = 4ax is a I ti 68. The length of the focal chord through the point t on the parabola y 2 = 4ax is a(t + l/t) . 2 69. If the normal at ti on the parabola y = 4ax meets it again at t2 then t2 70. If the normal at t on the parabola y = 4ax subtends a right angle i) at its focus then t = ± 2 ii) at its vertex then t = 71. The orthocentre of the triangle formed by three tangents of a parabola lies on the directrix
  5. 72. The angle between the pair of tangents drawn from (Xl, YD to the parabola S Sll tan y — 4ax — — 0 is TABLES FORM OF CONIC SECTION : (FORMULAS) PARABOLA S.N o i) Equation Y =4ax (y—k)2=4a(x— h) Vertex Focus Latus rectum 4 lal 4 lal 4 lal Axis 0 iii) Equation b2 a 2 2 x 2 b2 a Equation k) Centre Centre Horizontal ellipse (a > b) e = x a2 -b2 a Focii Directricies Min Major axis a -b2,0) ( a2 —b2 )x = +a2 or axis o 2 b2 Vertical ellipse (a > b) e = ( b2 -a2)y b2 Focii Hyperbola e Directricies b2 2 a Major axis Tangent at vertex 2 a Latus rectum b2 a 2 b Latus Directix a=0 Verticies Verticie s 2 x a 2 Minor rectu axis b2 a Equation of L.R. x—a=0 x—h—a = y—k—a = Property SP+SIP =2a SP+SIP Property Isp-slpl =2a The equation of the tangent at (Xl, YD , the equation of the chord of contact of (XI,YI) and polar of (Xl, YD with respect to S = 0 is Sl = 0.
  6. Curve Parabola Ellipse Hyperbola Parametric equations : Curve Parabola Ellipse Hyperbola Parabola Equation of the tangent, chord of contact, and the polar at (Xl, Yl) 2, = 2at —seco - —tan 0 = 1 Equation 2 = 4ax 2 2 b2 yyl= 2ax + 2aX1 2 a Equation Y =4ax 2 2 2 b2 a 2 a b2 b2 a 2 a b 2 b2 Parametric point t = (at2, 2at) 0 = (acos0, bsin0) 0 = (asec0, btan0) Parametric equation x = at y x = acos0, y = bsin0 x = asec0 y , = btan0 Equation of the chord joining two parametric points : Curve Parabola Ellipse Hyperbola Equation Y =4ax 2 2 Point ti t2 Equation of the chord (ti + t2)y —2x =2at1t2 x b2 2 b2 + — Sin —cos b 2 2 a = cos 2 a 2 a 01 + x + — Sin —cos b 2 a Equation of the tangent x —cos0 +—sin0 = 1 2 = cos Slope 1 —b sin 0 asin0 b a sin 0 2 Equation of the tangent at the parametric point : Curve Parabola Ellipse Hyperbola Equation Y =4ax 2 2 Point t a 2 a b2 2 b a x a b b
  7. Equation of the normal at the parametric point : Curve Parabola Ellipse Hyperbola Equation Y = 4ax 2 2 Point t Equation of the Normal Y + tx = 2at + at-I-3 ax cos 0 ax by sin 0 by -b2 2 a 2 a b2 2 b2 cos0 sino Parabola Slope —t a sin 0 b cos 0 -a sin 0 b Condition for tangency and the point of contact (y = mx + c) : Curve Parabola Ellipse Hyperbola Curve Parabola Parabola Ellipse Hyperbola Curve Parabola Ellipse Hyperbola Equation Y =4ax 2 Condition for tangency Point of contact a m 2 = am + b c m a 2 a b2 b2 c c 2a or m c c a m b2 c —b2 c 2a m Condition that he line Ix + my + n = 0 is a tangent . Equation Condition of tangency 2 2 x 2 a 2 a = 4ax 2 b2 2 b2 2 In = am 2 mn = al a212+bm— 22 —b m — al 2 2 Equation of the tangent is of form . Equation 2 = 4ax 2 Equation of the tangent a Y=mx + m a 2 a b2 2 y = mx + mx ± -b2
  8. Curve Parabola Ellipse Hyperbola Parabola Equation from which the slopes of the tangents through (XI,YI) are given — — 2aX1 Equation Condition of tangency m — myl+ a —a ) — 2x1Y1m +YI b — a2) — 2x1Y1m -FYI + b a = 4ax b2 The equation of the chord having the mid point (Xl, YD is Equation a2 b2 The condition that the lines lix + nuy + m = 0 and 12x + + 112= 0 may be conjugate with respect to Equation Curve Parabola Ellipse Hyperbola Curve Parabola Ellipse Hyperbola Equation of the chord YYI -2aXl - XXI Yl Yl Condition a 11 12 + b num2 = a 11 12 —b minu — a = 4ax b2 The locus of the point of intersection of the perpendicular tangents is Curve Parabola Ellipse Hyperbola Equation Equation of the locus Directrix — x + a Director circle — Director circle a b2
  9. For any conic, the tangents at the end of Latus rectum, the corresponding directrix and the axis are Parabola The locus of the points whose chords of contact subtend a right angle at the origin is (a, —2a) Curve Parabola Ellipse Hyperbola Curve Parabola Ellipse Hyperbola Equation a = 4ax b2 Ends of latus rectum . Equation Equation of the locus x + 4a = 0 Equation of the locus (a, 2a), (a, — 2a) a b2 Curve Parabola Ellipse Hyperbola Equation concurrent. (a, 2a) Point of concurrency a a = 4ax b2 b2

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