Find The Locus Of The Mid-point Of The Chords Of The Parabola y2 = 4ax which Subtend A Right Angle At The Vertex Of The Parabola.

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


Correct option is

y2 – 2ax + 8a2 = 0

Let P(h, k) be the mid-point of a chord QR of the parabola

y2 = 4ax, then equation of chord QR is

               T = S1

or     yk – 2a(x + h) = k2 – 4ah

     yk – 2ax = k2 – 2ah

Let A is vertex of the parabola. For combined equation of AQand AR, use homogenization of y2 = 4ax with the help of (1).


    y2(k2 – 2ah) – 4akxy +8a2x2 = 0


 Coefficient of x2 + coefficient of y2 = 0

        k2 – 2ah + 8a2 = 0

Hence the locus of P(h, k) is

          y2 – 2ax + 8a2 = 0





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