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  • The locus of the mid points of the chords of the parabola y2 = 8x which subtend a right angle at the vertex of the parabola isOption: 1 <i

The locus of the mid points of the chords of the parabola y2 = 8x which subtend a right angle at the vertex of the parabola is

Option: 1

y^{2}+4x-32=0


Option: 2

y^{2}-4x+32=0


Option: 3

y^{2}+4x+32=0


Option: 4

y^{2}-4x-32=0


Answers (1)

best_answer

Let P (h, k) be the mid point of a chord QR of the parabola y2 = 8x, then the equation of chord QR is

\\\mathrm{\;\;\;\;\;\;\;}T=S_{1} \Rightarrow y k-4(x+h)=k^{2}-8 h\\\Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;y k-4 x=k^{2}-4h \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(i)\\

If A is the vertex of the parabola. For the combined equation of AQ and AR, Making homogenous of y2 = 8x with the help of eq (i)

\\ {\therefore y^{2}=8 x \cdot 1} \\ {\therefore y^{2}=8 x\left(\frac{y k-4 x}{k^{2}-4 h}\right)} \\ {\Rightarrow y^{2}\left(k^{2}-4 h\right)-8 k x y+32 x^{2}=0}\\\text { since } \angle Q A R=90^{\circ}\\\begin{array}{l}{\therefore \text { Co-efficient of } x^{2}+\text { Co-efficient of } y^{2}=0} \\ {\Rightarrow k^{2}-4 h+32=0}\end{array}

Hence the locus is P (h, k) is y^{2}-4x+32=0

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