Question

A rod $PQ$ of length $2a$ slides with its ends on the axes then locus of circumcentre of $△OPQ$ is

• A. $x^2+y^2=2a^2$
• B. $x^2+y^2=4a^2$
• C. $x^2+y^2=3a^2$
• D. $x^2+y^2=a^2$

Answer 1

The correct option is D $x^2+y^2=a^2$

$\Delta OPQ$ is a right triangle

Circumcentre of right triangle is mid point oh Hypotenuse

$pa=2a$

M is the mid point of $PQ$

$\begin{array}{c}M\left(\frac{a+0}{2},\frac{0+b}{2}\right)=M\left(\frac{a}{2},\frac{b}{2}\right)\\ M\left(h,k\right)\end{array}$

$h=\frac{a}{2}$ , $k=\frac{b}{2}$

$a=2h$, $b = 2k$

$\begin{array}{c}P{a}^2=O{P}^2+O{Q}^2\\ y{a}^2=b^2+a^2\end{array}$

$\begin{array}{c}4a^2=4k^2+4h^2\\ h^2=k^2=a^2\end{array}$

So the locus of circumcentre of $\Delta OPQ$ is $x^2+y^2=a^2$