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. $4x^2+y^2-4kx=0$
• 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 a point on the parabola be
P $(a{t}^2,2at)$
Then, according to the question,
$SP=\sqrt{(a{t}^2-a{)}^2+(2at{)}^2}\Rightarrow k=\sqrt{(a{t}^2+a{)}^2}\Rightarrow k=a{t}^2+a\cdots \left(1\right)$

Let $(\alpha ,\beta )$ be the moving point. Then,
$\alpha =a{t}^2,\beta =2at$
$\Rightarrow \frac{\alpha }{\beta }=\frac{t}{2}$
$a=\frac{{\beta }^2}{4\alpha }$
$[\because \text{point}(\alpha ,\beta )\text{lies on}{y}^2=4ax]$
On substituting these values in (1), we get
$a{t}^2+a=k\Rightarrow \alpha +\frac{{\beta }^2}{4\alpha }=k$
$\Rightarrow {\beta }^2+4{\alpha }^2=4k\alpha$

Hence the locus is,
$4x^2+y^2-4kx=0$