The tangent at the point P(x1y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, the coordinates of the mid-point of QR are  


Correct option is


Equation of the tangent at P(x1y1) to the parabola y2 = 4ax is

                          yy1 = 2a(x + x1)

or                     2ax – y1y + 2ax1 = 0                      (1)

If M(h, k) is the mid-point of QR, then equation of QR a chord of the parabola y2 = 4a(x + b) in terms of its mid-point is

ky – 2a(x +h) – 4ab = k2 – 4a(h + b)

(using T = S’) or 2ax – ky + k2 – 2ah = 0              (2)

Since (1) and (2) represent the same line, we have


⇒            k = y1 and k2 – 2ah = 2ax1

⇒   y12 – 2ah = 2ax1­   ⇒ 4ax1 – 2ax­1 = 2ah

⇒                 h = x1



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