Question

The equation of the locus of a point which moves, so that its distance from the point $(ak,0)$ is $k$ times its distance from the point $(\frac{a}{k},0),(k\ne 1)$ is

• A. $x^2-y^2=a^2$
• B. $2x^2-y^2=2a^2$
• C. $xy=a^2$
• D. $x^2+y^2=a^2$

Answer 1

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

Let the point on locus be $(x,y)$

$\therefore$ ${(x-ak)}^2+y^2=k^2[{(x-a/k)}^2+y^2]$

$\Rightarrow x^2+a^2{k}^2-2axk+y^2=k^2[x^2+\frac{a^2}{k^2}-2\frac{ax}{k}+y^2]$

$\Rightarrow x^2+y^2=k^2[x^2+y^2]-a^2{k}^2+a^2$

$\Rightarrow [x^2+y^2][k^2-1]=a^2[k^2-1]$

$\Rightarrow x^2+y^2=a^2$ [D]