Question

A circle drawn on any focal chord AB of the parabola y2 = 4axas diameter cuts the parabola again at and D. If the parameters of the points A, B, C, D be t1, t2 t3 and t1 respectively, then the value of t3 t4 is

Solution

Correct option is

3

Since AB is a focal chord,                  … (1)

Circle on AB as diameter is

(x – at12)(x – at22) + (y – 2at1)(– 2at2) = 0

 x2 + y2 – ax(t12 + t22) – 2ay(t1 + t2) + a2t12t22 + 4a2t1t2 = 0

It meets the parabola y2 = 4ax, i.e., x = aT2= 2aT in four points A, B, C, D

a2T4 + 4a2T2 – a2T2(t12 + t22) – 4a2T(t1 + t2) + a2(–1)2 + 4a2(–1)= 0

a2T4 + 0T3 + a2T2(4 – t12 – t22) – 4a2T(t1 + t2) – 3a2 = 0

It is a fourth degree equation in T whose roots are t1, t2 t3t4

   

   t3 t4 = 3.

SIMILAR QUESTIONS

Q1

A tangent and a normal are drawn at the point P(16, 16) of the parabola y2 = 16x which cut the axis of the parabola at the points A and B respectively. If the center of the circle through P, A and B is C, then angle between PC and axis of x is

Q2

If x + y = k is normal to y2 = 12x, then k is

Q3

The length of normal chord which subtends an angle of 900 at the vertex of the parabola y2 = 4x is

Q4

A focal chord of parabola y2 = 4x is inclined at an angle of  with the +ive direction of x-axis, then the slope of normal drawn at the ends of focal chord will satisfy the equation

Q5

If two different tangents of y2 = 4x are the normal’s to the parabola x2 = 4ay, then

Q6

Find the locus of the mid-points of the chord of the parabola y2 = 4ax which subtend a right angle at the vertex.

Q7

If the parabola C and D intersect at a point A on the line L1, then equation of the tangent point L at A to the parabola D is

Q8

If a > 0, the angle subtended by the chord AB at the vertex of the parabola is

Q9

P is a point on the circle C, the perpendicular PQ to the major axis of the ellipse E meets the ellipse at M, then  is equal to

Q10

Equation of the diameter of the ellipse conjugate to the diameter respected by L is