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Chapter 12

Chapter 12. THE PARABOLA. 抛物线. Definition:. A parabola is defined as the locus of a point which moves so that its distance from a fixed point is always equal to its distance from a fixed line. a. x. y. S is called the focus. M. P(x,y). O. x. (-a,0). S(a,0).

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Chapter 12

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  1. Chapter 12 THE PARABOLA 抛物线 Parabola

  2. Parabola

  3. Definition: A parabola is defined as the locus of a point which moves so that its distance from a fixed point is always equal to its distance from a fixed line. Parabola

  4. a x y S is called the focus. M P(x,y) O x (-a,0) S(a,0) The fixed line, x=-a is called the directrix of the parabola. x=-a O is called the vertex of the parabola. Parabola

  5. Based on the definition, PS=PM This is the equation of the parabola. Parabola

  6. y y directrix is x=-a focus is (-a,0) x x 0 0 focus is (a,0) directrix is x=a Parabola

  7. y y 0 x focus is (0,a) focus is (0,-a) x 0 Parabola

  8. y l F x O y l O F x y F O x l y l x O F x≥0 y∈R y2 = 4ax （a>0） x=-a y2 = -4ax （a>0） y≥0 x∈R x2 = 4ay （a>0） y ≤0 x∈R x2 = -4ay （a>0）

  9. General parabola The general form of a parabola is : Which is derived from the general conic equation and the fact that, for a parabola Parabola

  10. e.g. 1 Parabola

  11. e.g. 2 p.156 Ex12a (8) Parabola

  12. e.g. 3 p.156 Q 19 Parabola

  13. Parabola

  14. e.g. 4 Find the equation of the parabola with focus (2,1) and directrix x+y=2. Parabola

  15. Parabola

  16. Parabola

  17. Equation of the tangent at (x’,y’) to the parabola Parabola

  18. Differentiating w.r.t x , Gradient of tangent at point (x’,y’)=2a/y’ Parabola

  19. Equation of tangent is As (x’,y’) lies on the curve, Parabola

  20. e.g. 5 Find the point of intersection of the tangent at the point (2,-4) to the parabola and the directrix. Given that the parabola is . Soln : Comparing with the standard eqn. We have a=2 . Eqn of tangent at (2,-4) is y(-4)=4(x+2) x+y+2=0 . Parabola

  21. Parametric equations of a parabola Parabola

  22. The equation is always satisfied by the values The parametric coordinates of any point on the curve are . Parabola

  23. e.g. 6 Find the parametric equations of the parabola Soln : (i) a=3, (ii) a=-3, (iii) a=1, Parabola

  24. Focal chords Parabola

  25. A chord of a parabola is a straight line joining any two points on it and passing thru’ the focus S. Y F X O Parabola

  26. The focal chord perpendicular to the axis of the parabola is called the latus rectum. Half the latus rectum is the semi-latus rectum. Parabola

  27. e.g. 6 Find the length of the latus rectum of the locus . Soln: Focus is (6,0) When x=6, y=12 or -12 Hence, latus rectum=24. Parabola

  28. e.g. 7 A focal chord is drawn thru’ the point on the parabola . Find the coordinates of the other end of the chord. Parabola

  29. Soln: Let the coordinates of Q be . y P Gradient of PF=gradient of FQ x F(a,0) 0 Q But n-t≠0 Parabola

  30. Hence, the coordinates of Q are . Note : The product of the parameters of the points at the extremities of a focal chord of a parabola is -1. What? Parabola

  31. Tangent and Normal at the point to the parabola . Parabola

  32. At the point , Equation of tangent at this point is : Parabola

  33. Gradient of normal at =-t Equation of normal is : Parabola

  34. Equation of a tangent in terms of its gradient Parabola

  35. The equation of the tangent at to the parabola , is Writing the gradient 1/t, as m, this equation becomes : i.e. Parabola

  36. We have Therefore , the point of contact of the tangent is . Parabola

  37. Remember this : For all values of m, the straight line is a tangent to the parabola . Parabola

  38. e.g. 8 Find the equations of the tangents from the point (2,3) to the parabola . Soln: We known, a=1 Parabola

  39. At (2,3) The tangents from the point (2,3) are : i.e. 2y=x+4 Parabola

  40. y=(1)x+1 i.e. y=x+1 Ans :2y=x+4 and y=x+1 Parabola

  41. Miscellaneous examples on the parabola Parabola

  42. e.g. 9 S is the focus of the parabola and P is the point (-3,8). PS meets the parabola at Q and R. Prove that Q, R divide PS internally and externally in the ratio 5:3. Parabola

  43. Soln: (-3,8) Q O S R Parabola

  44. Parabola

  45. Parabola

  46. e.g. 10 If the tangents at points P and Q on the parabola are perpendicular, find the locus of the midpoint of PQ. Parabola

  47. Soln: Gradient of tangent at P Gradient of tangent at Q Parabola

  48. If the mid-point of PQ is (x’,y’) then (1) (2) Square the (2), Parabola

  49. So, the locus of the midpoint of PQ is Parabola

  50. e.g. 11 Prove that the two tangents to the parabola , which pass thru’ the point (-a,k), are at right angles. Parabola

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