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Aline cuts the $x$ -axis at $A(7,0)$ and the $y$ -axis at $B(0,-5)$. A variable line $P Q$ is drawn perpendicular to $A B$ cutting the $x$ -axis at $P(a, 0)$ and the $y$ -axis at $Q(0, b)$. If $A Q$ and BP intersect at $R$, the locus of $R$ is
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The correct answer is:
$x^{2}+y^{2}-7 x+5 y=0$
Hint:
$\mathrm{P}$ is orthocentre of $\triangle \mathrm{ABQ}$
$\mathrm{m}_{\mathrm{BR}} \times \mathrm{m}_{\mathrm{AR}}=-1$
$\Rightarrow\left(\frac{\mathrm{k}+5}{\mathrm{~h}}\right) \times\left(\frac{\mathrm{k}}{\mathrm{h}-7}\right)=-1$
$\Rightarrow x^{2}+y^{2}-7 x+5 y=0$
$\mathrm{P}$ is orthocentre of $\triangle \mathrm{ABQ}$
$\mathrm{m}_{\mathrm{BR}} \times \mathrm{m}_{\mathrm{AR}}=-1$
$\Rightarrow\left(\frac{\mathrm{k}+5}{\mathrm{~h}}\right) \times\left(\frac{\mathrm{k}}{\mathrm{h}-7}\right)=-1$
$\Rightarrow x^{2}+y^{2}-7 x+5 y=0$
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