Question

Locus of image of the point $P(h,k)$ with respect to the line mirror which passes through the origin is

• A. $x^2+y^2=hx+ky$
• B. $x^2+y^2=h^2+k^2$
• C. $x^2+y^2=h^2+hx$
• D. $x^2+y^2=h^2+k^2+hx+ky$

Answer 1

The correct option is B $x^2+y^2=h^2+k^2$

Let the line be $y=mx$ and image of $P(h,k)$ be $Q(a,b)$
Then mid-point of $PQ$ will lie on $y=mx$
$\therefore \left(\frac{h+a}{2}\right)\cdot m=\frac{b+k}{2}$
$\Rightarrow m=\frac{b+k}{h+a}$

Also, line $PQ$ is perpendicular to line $y=mx$
$\Rightarrow \left(\frac{b+k}{h+a}\right)\cdot \left(\frac{b-k}{a-h}\right)=-1\Rightarrow b^2-k^2=h^2-a^2\Rightarrow a^2+b^2=h^2+k^2$

Hence locus will be $x^2+y^2=h^2+k^2$