Question

The locus of a point on the variable parabola $y^2=4ax$, whose distance from focus is always equal to $k$, is equal to ($a$ is parameter)

• A. $4x^2+y^2-4kx=0$
• B. $x^2+y^2-4kx=0$
• C. $2x^2+4y^2-8kx=0$
• D. $4x^2-y^2+4kx=0$

Answer 1

The correct option is A $4x^2+y^2-4kx=0$

Let $P(\alpha ,\beta )$ be a point on the parabola $y^2=4ax$.

Then according to question, $SP=\alpha +\beta =k$ ....(i)

Since ($\alpha ,\beta$) lies on $y^2=4ax$

${\beta }^2=4a\alpha$
$\Rightarrow a=\frac{{\beta }^2}{4\alpha }$

Substitute the values in equation (i),
$\alpha +\frac{{\beta }^2}{4\alpha }=k$

$\Rightarrow {\beta }^2+4{\alpha }^2=4k\alpha$
$\Rightarrow 4x^2+y^2-4kx=0$ is the required locus.