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
$\because AB=AC$
$\Rightarrow \sqrt{{\left(1+1\right)}^2+{\left(2-0\right)}^2}$
$=\sqrt{{\left(-1+1\right)}^2+{\left(0-k\right)}^2}$
$\Rightarrow 4+4=k^2$
$\Rightarrow k^2=8$
$\Rightarrow k=2\sqrt{2}$