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

• A

  $x^{2}+y^{2}+7 x+5 y=0$
• B

  $x^{2}+y^{2}+7 x-5 y=0$
• Correct answer
  C

  $x^{2}+y^{2}-7 x+5 y=0$
• D

  $x^{2}+y^{2}-7 x-5 y=0$

Solution

Hint: $\mathrm{P}$ is orthocentre of $\triangle \mathrm{ABQ}$ $\mathrm{m}_{\mathrm{BR}} \times \mathrm{m}_{\mathrm{AR}}=-1$ $\Rightarrow\left(\frac{\mathrm{k}+5}{\mathrm{~h}}\right) \times\left(\frac{\mathrm{k}}{\mathrm{h}-7}\right)=-1$ $\Rightarrow x^{2}+y^{2}-7 x+5 y=0$

Hence, the correct answer is option: 'C'.