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Mathematics

Mathematics. Session . Parabola Session 3. Session Objective. Number of Normals Drawn From a Point Number of Tangents Drawn From a Point Director circle Equation of the Pair of Tangents Equation of Chord of Contact Equation of the Chord with middle point at (h, k)

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Mathematics

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  1. Mathematics

  2. Session Parabola Session 3

  3. Session Objective • Number of Normals Drawn From a Point • Number of Tangents Drawn From a Point • Director circle • Equation of the Pair of Tangents • Equation of Chord of Contact • Equation of the Chord with middle point at (h, k) • Diameter of the Parabola • Parabola y = ax2 + bx + c

  4. Number of Normals Drawn From a Point (h,k) Parabola be y2 = 4ax let the slope of the normal be m, then its equation is given by y = mx – 2am – am3 if it passes through (h,k) then k = mh – 2am – am3 i.e. am3 + (2a – h)m + k = 0 This shows from (h,k) there are three normals possible (real/imaginary) as we get cubic in m

  5. Observations from am3 + (2a – h)m + k = 0 • At least one of the normal is real as cubic equation have atleast one real root • The three feet of normals are called Co-Normal points given by (am12, –2am1 ), (am22, –2am2) and (am32, –2am3) where mi’s are the roots of the given cubic eqn • Sum of the ordinates of the co-normal points = –2a (m1 + m2 + m3) = 0 • Sum of slopes of normals at co-normal points = 0 • Centroid of triangle formed by co-normal points lies on axis of the parabola.

  6. Observations from am3 + (2a – h)m + k = 0 7. Thus we have following different cases arises: • 3 real and distinct roots m1, m2, m3 or m1, m2, –m1–m2 • 3 real in which 2 are equal m1, m2, m2 or –2m2, m2, m2 • 3 real, all equal m1, m1, m1 or 0, 0, 0  k = 0 , h = 2a • 1 real, 2 imaginary m1,   i (  0)

  7. Number of Tangents Drawn From a Point (h,k) Parabola be y2 = 4ax let the slope of the tangent be m, then its equation is given by y = mx + a/mif it passes through (h,k) then k = mh + a/m i.e. hm2 – km + a = 0 This shows from (h,k) there are two tangents possible (real/imaginary) as we get quadratic in m

  8. Observations from hm2 – km + a = 0 Discriminant = k2 – 4ah = S1 • S1 > 0 Point is outside parabola: 2 real & distinct tangents • S1 = 0 Point is on the parabola: Coincident tangents • S1 < 0 Point is inside parabola: No real tangent • m1 + m2 = k/h , m1m2 = a/h

  9. Director Circle Locus of the point of intersection of the perpendicular tangents is called Director Circle hm2 – km + a = 0 m1m2 = a/h = –1  h = –a i.e. locus is x = –a Hence in case of parabola perpendicular tangents intersect at its directrix. Director circle of a parabola is its directrix.

  10. Equation of the Pair of Tangents Parabola be y2 = 4ax then equation of pair of tangents drawn from (h,k) is given by SS1 = T2 where S  y2 – 4ax, S1  k2 – 4ah and T  ky – 2a(x + h) Pair of Tangents: (y2 – 4ax)(k2 – 4ah) = (ky – 2a(x + h))2

  11. Equation of Chord of Contact Parabola be y2 = 4ax then equation of chord of contact of tangents drawn from (h,k) is given by T = 0 where T  ky – 2a(x + h) Chord of Contact is: ky = 2a(x + h)

  12. Equation of the Chord with middle point at (h, k) Parabola be y2 = 4ax then equation of chord whose middle point is at (h,k) is given by T = S1 where T  ky – 2a(x + h) and S1  k2 – 4ah Chord with middle point at (h,k) is: ky – 2a(x + h) = k2 – 4ah i.e. ky – 2ax = k2 – 2ah

  13. Diameter of the Parabola Diameter: Locus of mid point of a system of parallel chords of a conic is known as diameter Let (h, k) be the mid point of a chord of slop e m then its equation is given by ky – 2ax = k2 – 2ah if its slope is m then Locus is

  14. Class Exercise

  15. Class Exercise - 1 Find the locus of the points from whichtwo of the three normals coincides.

  16. Let (h, k) be the point and m be the slope of thenormal then As two normal coincides Let m1 =m2then and Solution

  17. Solution contd..

  18. Three normals with slopes m1, m2and m3 are drawn from a point Pnot on the axis of the parabola y2 = 4x.If results in the locus of Pbeing a part of the parabola, find thevalue of Class Exercise - 2

  19. Let P be (h, k)then mi’s are given by Solution m3satisfies (i)

  20. locus of (h, k) is If it is a part of y2 = 4x Solution contd..

  21. Class Exercise - 3 Find the locus of the middle pointsof the normal chords of the parabolay2 = 4ax.

  22. Let (h, k) be the middle point then its equation is given by T = S1i.e. ky – 2a (x + h) = k2 – 4ah If it is also the normal of y2 = 4ax then compare it with Solution

  23. Solution contd..

  24. Class Exercise - 4 Find the locus of the point ofintersection of the tangents at theextremities of chord of y2 = 4ax whichsubtends right angle at its vertex.

  25. Pair of lines are given by homogenising y2 = 4ax using (i) Solution Let (h, k) be the point of intersection of tangents then its chord of contact is given by T = 0 i.e. ky – 2a (x + h) = 0 ...(i) Now according to the question pair of lines joining origin to the point of intersection of (i) with the parabola are at right angles.

  26. Let be the extremities of the chord. As chord subtends right angle at the vertex we have Solution contd.. 4ax2 – 2kxy + hy2 = 0 Pair of lines are perpendicularif 4a + h = 0 hence locus of (h, k) is x + 4a = 0 Alternative:

  27. Point of intersection of tangents at these point is point becomes (–4a, a (t1 + t2)) and its locus isx + 4a = 0 Solution contd..

  28. Class Exercise - 5 Find the equation of the diameter of theparabola given by 3y2 = 7x, whose systemofparallel chords are y = 2x + c.

  29. Its slope is 2 Solution Let (h, k) be the middle point of the chord then its equation is given by T = S1

  30. Solution contd.. Alternative:

  31. Three normals to the parabola y2 = xare drawn through a point (c, 0), then • (b) (c) (d) None of these Class Exercise - 6

  32. i.e. Solution above equation have 3 real roots if 2 – 4c < 0 Hence,answer is (c).

  33. Class Exercise - 7 The mid point of segmentintercepted by the parabola x2 =6yfrom the line x – y = 1 is ___.

  34. h = 3, k = 2 Solution Let the mid point be (h, k), therefore, its equation is given by hx – 3 (y + k) = h2 – 6k or hx – 3y = h2 – 3k Hence (3, 2) is the mid point of the segment.

  35. Class Exercise - 8 Draw y= –2x2 + 3x + 1.

  36. Solution

  37. Find the locus of the point of intersectionof tangents to y2 = 4ax which includes anangle between them. Class Exercise - 9

  38. Solution Let (h, k) be the point of intersection then equation of pair of tangents are given by SS1 = T2

  39. Solution contd..

  40. Class Exercise - 10 Find the coordinates of feet of thenormals drawn from (14, 7) to theparabola y2= 16x + 8y.

  41. Let the foot of the normal be then tangent at is given by Solution

  42. which also passes through (14, 7) Corresponding Feet of normals are(0, 0), (8, 16) and (3, –4) Solution contd..

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