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Two variable perpendicular chords are drawn from the origin \mathrm{' \mathrm{O}'} to the parabola \mathrm{y=x^2-x}, which meet the parabola at P and Q. Rectangle POQR is completed. Find the locus of R.

Option: 1

\mathrm{x^2+3 x-y+4=0}


Option: 2

\mathrm{x^2-3 x-y-4=0}


Option: 3

\mathrm{x^2-3 x-y+4=0}


Option: 4

\mathrm{\text { None of these }}


Answers (1)

\mathrm{\text { Let } R \equiv(h, k)}

\mathrm{\text { Mid point of } O R \equiv\left(\frac{\mathrm{h}}{2}, \frac{\mathrm{k}}{2}\right)}

\mathrm{\text { Given curve } S \equiv x^2-x y=0}

Equation of chord \mathrm{ \mathrm{PQ}} with mid point \mathrm{\left(\frac{\mathrm{h}}{2}, \frac{\mathrm{k}}{2}\right)} is

\mathrm{ \begin{aligned} & \frac{x h}{2}-\frac{1}{2}\left(x+\frac{h}{2}\right)-\frac{1}{2}\left(y+\frac{k}{2}\right)=\left(\frac{h}{2}\right)^2-\frac{h}{2}-\frac{k}{2} \\\\ & \Rightarrow \frac{2(h-1)}{h^2-h-k} x-\left(\frac{2}{h^2-h-k}\right) y=1 \end{aligned} }
Making the equation of the parabola homogeneous with the help of (1), we have            ....(1)

\mathrm{ x^2-x\left[\frac{2(h-1)}{h^2-h-k} x-\left(\frac{2}{h^2-h-k}\right) y\right]-y\left[\frac{2(h-1)}{h^2-h-k} x-\left(\frac{2}{h^2-h-k}\right) y\right]=0 }

Since angle between the lines given by (2) is \mathrm{ \pi / 2}.                                               ....(2)

\mathrm{\begin{aligned} & \quad 1-\frac{2(h-1)}{h^2-h-k}+\frac{2}{h^2-h-k}=0 \\\\ & \therefore \quad h^2-h-k 2(h 1)+2=0 \\\\ & \Rightarrow h^2-3 h-k+4=0 \\\\ & \text { Required locus is } x^2-3 x y+4=0 . \end{aligned}}

Posted by

Ramraj Saini

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