Question

A line passing through $P(6,4)$ meets the coordinates axes at $A$ and $B$ respectively. If $O$ is the origin, then the locus of the centre of the circumcircle of triangle $OAB$ is

• A. $x^{-1}+2y^{-1}=1$
• B. $3x^{-1}+y^{-1}=1$
• C. $x^{-1}+y^{-1}=1$
• D. $3x^{-1}+2y^{-1}=1$

Answer 1

The correct option is D $3x^{-1}+2y^{-1}=1$

Let $\frac{x}{a}+\frac{y}{b}=1$ is the line passing through $(6,4)$ and $(h,k)$ is the circumcentre
$\Rightarrow \frac{6}{a}+\frac{4}{b}=1\cdots \left(1\right)$
$\because △OAB$ is a right angle triangle, circumcentre will be the mid point of $AB$
$\Rightarrow h=\frac{a}{2}\Rightarrow a=2h$
and $k=\frac{b}{2}\Rightarrow b=2k$
putting these values in $\left(1\right)$
$\Rightarrow \frac{6}{2h}+\frac{4}{2k}=1\Rightarrow \frac{3}{h}+\frac{2}{k}=1$
$\Rightarrow$Locus : $3x^{-1}+2y^{-1}=1$