Question

If $M$ is the foot of the perpendicular drawn from the origin $O$ on to the variable line L,passing through a fixed point $(a, b)$ then the locus of the mid point of $OM$ is

• A. $x^{2}+y^{2}=a^{2}+b^{2}$
• B. $2 x^{2}+2 y^{2}-a x-b y=0$
• C. $a x+b y=0$
• D. $2 x^{2}+2 y^{2}-a y-b x=0$

Answer 1

Answer: (B)

We have, $R(h, k)$ is mid-point of $O M$.
$\therefore \left(\frac{0+\alpha}{2}, \frac{0+\beta}{2}\right)=(h, k)$
$ \Rightarrow \alpha=2 h, \beta=2 k$
$\Rightarrow $ Coordinates of $M$ are $(2 h, 2 k)$.
Now, slope of $O M=\frac{2 k-0}{2 h-0}=\frac{k}{h}$
and slope of $M Q=\frac{2 k-b}{2 h-a}$
Since, $O M \perp M Q$
$\therefore \frac{k}{h} \times \frac{2 k-b}{2 h-a}=-1$
$ \Rightarrow 2 k^{2}-b k=-2 h^{2}+a h$
$\Rightarrow 2 h^{2}+2 k^{2}-a h-b k=0$
$\therefore $ Locus of $R(h, k)$ is
$2 x^{2}+2 y^{2}-a x-b y=0$