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Question: If the tangent at the point P(x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at ...

If the tangent at the point P(x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, then the mid-point of QR is-

A

(x1 + b, y1 + b)

B

(x1 – b, y1 – b)

C

(x1, y1)

D

(x1 + b, y1)

Answer

(x1, y1)

Explanation

Solution

Equation of the tangent at P(x1, y1) to y2 = 4ax is

yy1 – 2ax – 2ax1 = 0.... (1)

Equation of the chord of y2 = 4a(x + b) whose mid-point is (x¢, y¢) is

yy¢ – 2ax – 2ax¢ – 4ab = y¢2 – 4ax¢ – 4ab

(i.e.) yy¢ – 2ax – (y¢2 – 2ax¢) = 0

Equations (1) and (2) represent the same line

\ y1y′=2a2a=2ax1y′2–2ax′\frac{y_{1}}{y'} = \frac{2a}{2a} = \frac{2ax_{1}}{y'^{2}–2ax'}

This gives y¢ = y1 and then 2ax1 = y¢2 – 2ax¢

= y12 – 2ax¢

= 4ax1 – 2ax¢

\ x¢ = x1

\ mid-point (x¢, y¢) = (x1, y1).