The tangent at the point P(x 1 , y 1 ) to the parabola y 2 = 4ax meets the parabola y 2 = 4a(x + b) at Q and R, the coordinates of the mid-point of QR are
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Equation of the tangents at P(x 1 , y 1 ) to the parabola y 2 = 4ax is yy 1 = 2a(x + x 1 )
or 2ax – y 1 y + 2ax 1 = 0 ….(i)
If M(h, k) is the mid-point of QR, then equation of QR a chord of the parabola y 2 = 4a(x + b) in term of its mid-point is ky – 2a (x + h) – 4ab
= k 2 – 4a (h + b) (using T = S ′ )
or 2ax – ky + k 2 – 2ah = 0 …...(ii)
Since (i) and (ii) represent the same line, we have
= 
= 
⇒ k = y 1 and k 2 – 2ah = 2ax 1
⇒ 
– 2ah = 2ax 1 ⇒ 4ax 1 – 2ax 1 = 2ah
⇒ h = x 1
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