CONIC SECTION - Lecture Notes

CONIC SECTION A conic is the locus of a point which keeps a constant ratio of the distance from a fixed point to the distance from a fixed line The fixed point is focus The fixed line is directrix The fixed ratio SP e  is called eccentricity PM If e  1  parabola If e  1  ellipse If e  1  hyperbola Qn. Mind the equation of the conic whose focus is (2,–3) directrix is x  y – 1  0 and having eccentricity 2 Ans. Let P  h, k be any point on the conic SP  2 PM 1

SP = h  22  k  32 h  k 1 PM  2 SP  e PM h  22  k  32  h  k 1 2 2 on squaring both sides h  22  k  32  h  k 12 h2  4h  4  k2  9  6k  h2  k2 1 2hk  2k  2h 4h  6k 13  2hk  2k  2h 1 2h  8k  2hk 12  0 h  4k  hk  6  0 h  4k  hk  6  0 Putting h = x, k = y Locus  x  4y  xy  6  0 (This is a hyperbola  e  2  1.4 ) The line passing through the focus and perpendicular to the directrix is called axis The point of intersection of the axis and the conic is called vertex The chord of the conic passing through the focus are called focal chords The chords r to the axis are called double ordinates. Parabola The focal chord which is also a double ordinate is called latus rectum A parabola is the locus of a point which is equidistant from a fixed point called focus and a fixed line called directrix. Thus from the vertex, distance to its focus and directrix are always equal. This distance is represented as a 2

Standard Parabolas Case I Equation  y2  4ax, a  0 vertex  0, 0 focus  a, 0 directrix  x  a Axis  x axis  y  0 Latus rectum, equation  x  a length = 4a focal distance (SP) = x+a Case II 3

Replace all x as –x Equation y2  4ax, a  0 Vertex (0,0) Focus (–a,0) directrix x = a Axis : x-axis (y=0) L.R: equation : x = –a Length = 4a Focal distance SP  x  a Case III Equation x2  4ay, a  0 Vertex (0,0) Focus (0,a) Directx y = -a Axis y-axis (x=0) L.R equation y = a Length 4a Focal distance (SP) = y + a 4

Case IV Replace all y or –y Equation x2  4ay, a  0 Vertex (0,0) Focus (0, –a) Directrix y = a Axis Y-axis (x=0) L.R. equation : y = –a Length : 4a Focal distance (SP) = y  a Shifted parabolas 5

Equation  y  k 2  4a  x  h  a  0 Vertex (h,k) Focus a  h, k directrix x  h  a x  a  h Axis : y  k L.R  4a Case II  y  k 2  4a x  h, a  0 Case III  x  h2  4a  y  k , a  0 Case IV 6

x  h 2  4a  y  k , a  0 Qn. Find the vertex, focus, directrix, axis and length of latus rectum of the parabola x2  2x  4y 1  0 Ans: x2  2x  4y 1  0 x2  2x  4y 1 Completing the sq method (sq of half of coefficient of x) x2  2x 1  4y 11 x 12  4y x 12  4  y  0  x  h 2  4a  y  k  Vertex h, k  1, 0 Focus 0  h, a  k 1, 1 0 1, 1 directrix  y  a  k y 10 y 1 L.R  4a  4 Qn: Find vertex, focus, directrix, axis, latus rectum of the parabola y2  4y  x 1  0 Ans: y2  4y  x 1 y2  4y  4  x 1 4  y  22  x  3 y  22  x  3  y  k 2  4a  x  h  k  2, h  3 4a  1; a  1 4 Vertex h, k  3, 2 7

Focus  a  h, k  1  3, 2   11 , 2  4  4  directrix  x  h  a x3 1 4 x  3  1  13 44 LR = 1 Axis y  0  k  2 Parametric Equations of parabola  For the parabola y2  4ax, A general point on it can be written as at2 , 2at for any t  R thus x  at2 y  2at is known as the parametric equation of the parabola IIIly for the other parabolas, parametric points Parabolas Parametric point y2  4ax y2  4ax at2, 2at x2  4ay at2, 2at x2  4ay 2at, at2  2at, at2   y  k2  4a x  k  x  at2  h, y  2ab  k Tangents of the parabola Point form To determine the equation of the tangent to the parabola through point (x1,y1) on it, replace x2  xx1 8

y2  yy1 x  x  x1 2 y  y  y1 2 xy  xy1  yx1 2 Tangent at  x1, y1  yy1  4a  x  x1   2  yy1  2a x  x1  Qn. Find the equation of the tangent through the point (1,–2) to the parabola x2  4x  y  5  0 Ans:  x1, y1   1, 2 (This point satisfies the parabola) xx1  4  x  x1   y  y1  5  0  2  2 x  x 1  y  2  5  0 2 2x  4x  4  y  2 10  0 2x  y  4  0 9

Parametric form Tangent at  x1, y1  Parametric point Tangent yy1  2a x  x1  yt  x  at2 at2, 2at yt  x  at2  x1, y1   at2 , 2at at2, 2at xt  y  at2   y.2at  2a x  at2 2at, at2  xt  y  at2 2at, at2  yt  x  at2 at2  h, 2at  k   y  k t  x  h  at2 IIIly for the other standard parabolas, Parabola y2  4ax y2  4ax x2  4ay x2  4ay y  k 2  4a x  h  Slope form Slope of tangent = m 10

Equation of tangent y = mx + c  (1) y2 =4ax  2 Solving 1) & (2) mx  c2  4ax m2x2  c2  2mxc  4ax m2x2  2x mc  2a  c2  0  roots are equal B2  4AC  0 4mc  2a 2  14m2c2  m2c2  4a2  4amc  m2c2 4a2  4amc c a m  y  mx  c  y  mx  a with slope m m OR yt  x  at2   tangent at at2 ,2at 11

or y= x  at t Comparing with y  mx  c 1  m or t  1 tm Putting t  1 , y  mx a mm  Point of contact at2, 2at Point of contact   a , 2a   m2 m  IIIly for the other parabolas Parabola Tangent with slope m Point of contact y2  4ax y  mx  a  a , 2a  m  m2 m  y2  4ax y  mx  a  a , 2a  m  m2 m  x2  4ay x2  4ay y  mx  am2 2am,am2  y  mx  am2  y  k 2  4a x  h  2am, am2  yk xhm a  a  h, 2a  k  m  m2 m  Normals of parabola The line r to the tangent passing through the same point of contact is called normal 12

Equations of normals Case I (point form) Tangent at  x1, y1  yy1  2a x  x1  y  2a x  2a x1 y1 y1 y  mx  c m of tangent  2a y1 slope of normal  y1 2a  Equation of normal at x1, y1   y  y1   y1  x  x1  2a Case II (Parametre form) Normal at (x1, y1) y  y1  y1 x  x1  2a  x1, y1   at2 , 2at 13

  y  2at  2at x  at2 Parametric point Normal 2a at2, 2at y  xt  2at  at3 y  xt  2at  at3 IIIly for other parabolas at2, 2at y  xt  2at  at3 Parabola 2at,at2  x  yt  2at  at3 y2  4ax y2  4ax 2at, at2  x  yt  2at  at3 x2  4ay x2  4ay  at2  h, 2at  k  y  k  x  h t  2at  at3  y  k2  4a x  h Case III (Slope form)  for y2  4ax , normal at at2, 2at Normal  y  xt  2at  at3 y  tx  2at  at3 t  m t  m 14

Normal with slope m y  mx  2am  am3    Point of contact, at2 , 2at  am2 , 2am IIIly for other parabolas Parabola Normal with slope m Point at which normalis drawn y2 =4ax y  mx  2am  am3 y2 =-4ax am2 , 2am x 2 =4ay y = mx+2am+am3 am2, 2am x 2 =-4ay y  mx  2a  a  2a , a   y-k2 =4a x-h m2  m m2  Some important properties y  mx  2a  a  2a , a  m2  m m2   y  k  m x  h  2am  am3 am2  h, 2am  k    If the chord joining the points at12 , 2at1 the at22 , 2at2 is a focal chord, then t1t2  1 or  t2  1 t1 Thus  Thus if one end of the focal chord is  a , 2a  at2, 2at then the other end will be  t2 t  2. If AB is a focal chord, then the harmonic mean of the segments SA & SB is always semi-litus rectum i.e 2SA.SB  2a where S is the focus SA  SB  3. The length of focal chord of the parabola through the point at2 , 2at is, AB  a  t  1 2  t  15

4. The point of intersection of the tangents to the parabola y2  4ax through the points    at2, 2at&at2, 2 2at 2  y2  4ax is C at1t2, a t1  t2  5. Thus the tangents to the parabola through the end points of any focal chord always intersect atthe directrix   t1t2  1  C a, a t1  t2 which is on x = –a the directrix 16

6. Director circle (orthoptic locus) It is the locus of the points of intersection of the perpendicular tangents to a parabola. It is a straight line which is the directrix of the parabola 7. Thus the tangents drawn to the parabola through the end points of any focal chord are always perpendicular and hence they meet at the directrix    8. If the normal to the parabola at at12 , 2at1 intersects the parabola again at a point at22, 2at2 , then t2  t1  2 t1 17

   9. If the normals drawn at the points at12 , 2at2 and at12 , 2at2 intersect at the point on the parabola then, t1t2  2  Area enclosed by the parabolas y2  4ax and x2  4by A  16ab 3 Co-Normal Points We can draw a maximum of three normals to the parabola from any point in the xy plane. The points of intersection of three normals with the parabola are called co-normal points 18

11. Chord of contact Equation of chord of contact from P is (AB) T1 : yy1  2a x  x1   0 12. Chord bisecting at a given point Equation of chord AB bisecting at (x1,y1) is T1  S1 yy1  2a  x  x1   y12  4ax1 S1  y12  4ax1 If S1  0   x1, y1  lies on the parabola If S1  0  x1, y1  lies inside the parabola If S1  0   x1, y1  lies outside the parabola OA  OA  a 19

Ellipse: An ellipse is the locus of a point which is SP  e PM with e 1 It is also defined as the locus of a point in which the sum of the distances from two fixed points is always a constant SA  e, SA  e Ax Az SA  eAz & SA  eAZ SA  SA  eAZ  AZ 2a  eOZ  OA  OA  OZ 2a  e 20z  OZ  a e  Directrix x  a e SA  3A  eAZ  AZ SS  eOA  OZ  OZ  OA SS  2ae 20

OS  ae Focus  ae, o SP  e pm x  ae2  y2  e  x  a / e 1 x2  2.ex  a2e2  y2  xe  a 2  x 2e2  2aex  a 2    x2 1 e2  y2  a2 1 e2  x2  y2  a2 a2 1 e2 1 a2 1 e2  b2 x2 y2 e2  a2  b2   1, a2 a2 b2 e  a2  b2 a Standard Ellipses  Equation:x2 y2 a2   1, a  b, b2  a2 1 e2 b2 Centre = (0,0) vertices = a, 0 Foci = a, e, 0  c, 0 where ae  c  a2  b2 21

e  c  a2  b2 aa Directrix : x   a e Major axis : x-axis, length = 2a Minor axis : y- axis, length = 2b Latus rectum : Equation : x  c 2b2 Length = a SP  a  ex SP  a  ex SP  SP  2a Case II  Equation x2 y2 : b2   a  b, b2  a2 1 e2 a2 Centre = (0,0) Vertices = 0, ae  0, c when c  a2  b2 e c a Directrices : y   a e Major axis = y-axis, length = 2a 22

Minor axis = x-axis, length = 2b Latus rectum : Equation : y  c 2b2 Length = a SP  a  ey,S' P  a  ey,SP  S' P  2a Shifted Ellipses x  h2  y  k2 x  h2 y  k2 a2  b2  1 b2  a2  1 ab ab Parametric Equations 23

For the Ellipse x2  y2  1, x a cos , y  b sin  are the parametric equations. a2 b2 x2 b2 For   1, a  b b2 a2 x  b cos , y  a sin  are the parametric equations x  h2 y  k2 a 2  b2  1, a  b P.E are x  h  a cos , y  k  b sin  Equations of Tangents (1) Point form Tangent at (x1,y1) T1  0 xx1  yy1 1 a2 b2 (2) Parameter form Tangent to x2 y2  1 at a cos , b sin  a2  b2 x cos   y sin   1 ab (3) Slope form y  mx  c is a tangent to x2  y2 1 iff a2 b2 c2  a2m2  b2 or c   a2m2  b2 24

 Equation of tangent with slope ‘m’ is y  mx  a 2m2  b2 Points of contact =   a2m , b2   a2m2  b2   a2m2  b2  Normals (1) Point form Equation of normal to x2 y2 1 at  x1, y1  is a2  b2 a2x  b2y  a2  b2 x1 y1 (2) Parameter form Normal at a cos , b sin  is ax  by  a2  b2 cos  sin  (3) Slope form Equation of normal with shape m to x2  y2  1 is a2 b2  m a2  b2 y  mx  a2  b2m2 Director circle Locus of point of intersection of perpendicular tangents to the Ellipse It is a circle concentric with the ellipse and having radius a2  b2 Area of Ellipse Area of Ellipse is A  ab Hyperbola If e>1, The conic is called Hyperbola. It is also defined as the locus of a point in which the modulus of the difference of the distances from two fixed points keeps as a constant 25

x2 y2  a2 b2  Equation :  1, b 2  a 2 e2 1 Centre = (0,0) Vertices = a, 0 Foci = ae, 0  c, 0 where c  ae  a2  b2 c a2  b2 e  ,e 1 aa Directrices : x   a e Transverse Axis : X-axis, length = 2a Conjugate Axis : Y-axis, length = 2b Latus rectum : Equation : x  c 2b2 Length = a Focal distances, SP  ex  a S'P  ex  a ___________ SP  S'P  2a 26

Case II y2  x2 a2 b2  Equation :  1, b2  a 2 e2 1 Centre = (0,0) Vertices = 0, a Foci = 0, ae  0, c, c  a2  b2 e c,e1 a Directrices = y   a e 2b2 L.R = a S.P.=ey + a, S' P  ey  a SP  S'P  2a Shifted Hyperbolas x  h2 y  k2 y  k2 x  h2 a2  b2  1, & a2  b2  1 having centre = (h,k) and axes IIel to co-ordinate axes 27

Parametric Equations Parametric Equation x  a sec , y  b tan  Tangents (1) Point form Tangent at  x1, y1  to x2  y2 1 a2 b2 xx1  yy1 1 a2 b2 (2) Parameter form Tangent at (a sec , b tan ) is x sec   y tan   1 ab (3) Slope form y  mx  c is a tangent to x2  y2  1 iff a2 b2 c2  a2m2  b2 or C   a 2m2  b2 Hence equaiton of tangent is y  mx  a2m2  b2 Points of contact =   a2m , b2   a2m2  b2   a2m2  b2  28

Normals (1) Point form Equation of normal to x2 y2  1 at  x1, y1  is a2  b2 a2x  b2y  a2  b2 x1 y1 (2) Parameter form Equation of normal at a sec , b tan  is ax  by  a2  b2 sec  tan  (3) Equation of normal with slope ‘m’  m a2  b2 y  mx  a2  b2m2 Director circle x2 y2 For  1 a2 b2  Equation of director circle is x2  y2  a2  b2 Assymptotes Lines touching the Hyperbola at infinity are called assymptotes. Assymptotes always intersect at its centre For x2  y2 1 a2 b2 Equations of assymptotes are y   b x a Note : The equations of a Hyperbola and its assymptotes always differ only in the constant term Conjugate Hyperbola For the hyperbola x2  y2 1 a2 b2 The conjugate hyperbola is x2 y2   –1 a2 b2 29