question_answer

If a circle of radius R passes through the origin\[O\] and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from 0 on AB is:

A) \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=4{{R}^{2}}{{x}^{2}}{{y}^{2}}\]

B) \[{{({{x}^{2}}+{{y}^{2}})}^{3}}=4{{R}^{2}}{{x}^{2}}{{y}^{2}}\]

C) \[{{({{x}^{2}}+{{y}^{2}})}^{2}}=4{{R}^{2}}{{x}^{2}}{{y}^{2}}\]

D) \[({{x}^{2}}+{{y}^{2}})(x+y)={{R}^{2}}xy\]

Correct Answer: B

Solution :

Slope of \[AB=\frac{-h}{k}\]

Equation of AB is \[hx+ky={{h}^{2}}+{{k}^{2}}\]

\[A\left( \frac{{{h}^{2}}+{{k}^{2}}}{h},0 \right)B\left( 0,\frac{{{h}^{2}}+{{K}^{2}}}{k} \right)\]

As,        \[A=2R\]

\[\Rightarrow \]   \[({{h}^{2}}+{{k}^{2}})=4{{R}^{2}}{{h}^{2}}{{k}^{2}}\]\[\Rightarrow \]            \[({{x}^{2}}+{{y}^{2}})=4{{R}^{2}}{{x}^{2}}{{y}^{2}}.\]