Question:
If the common tangent to the parabolas, $y^{2}=4 x$ and $x^{2}=4 y$ also touches the circle, $x^{2}+y^{2}=c^{2}$, then $c$ is equal to:
Correct Option: , 2
Solution:
Equation tangent to parabola $y^{2}=4 x$ with slope $m$ be:
$y=m x+\frac{1}{m}$ ...(i)
$\because$ Equation of tangent to $x^{2}=4 y$ with slope $m$ be :
$y=m x-a m^{2}$ .......(ii)
From eq. (i) and (ii),
$\frac{1}{m}=-m^{2} \Rightarrow m=-1$
$\therefore$ Equation tangent $: x+y+1=0$
It is tangent to circle $x^{2}+y^{2}=c^{2}$
$\Rightarrow c=\frac{1}{\sqrt{2}}$