Question

A circle touches the parabola $y^{2}=4x$ at $(1,2)$ and also touches its directrix The $y$-coordinate of the point of contact of the circle and the directrix is

• A. $\sqrt{2}$
• B. $2$
• C. $2\sqrt{2}$
• D. $4$

Answer 1

Answer: (C)

We have,

Equation of parabola
$y^{2}=4x$
Equation of tangent of parabola at $(1, 2)$, is
$2y =4 \frac{(x+1)}{2}$
$\Rightarrow y=x+1$
The tangent of parabola $y=x+1$
intersect the directrix of parabola at $A(-1, 0)$
Now, $AB$ and $AC$ are tangent of circle
$\because AB = AC$
$\Rightarrow \sqrt{\left(1+1\right)^{2}+\left(2-0\right)^{2}}$
$=\sqrt{\left(-1+1\right)^{2}+\left(0-k\right)^{2}}$
$\Rightarrow 4+4=k^{2} $
$\Rightarrow k^{2}=8$
$\Rightarrow k=2\sqrt{2}$