If the common tangent to the parabolas,

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 :

  1. $\frac{1}{2}$

  2. $\frac{1}{2 \sqrt{2}}$

  3. $\frac{1}{\sqrt{2}}$

  4. $\frac{1}{4}$


Correct Option: , 3

Solution:

$y=m x+\frac{1}{m}\left(\operatorname{tangent}\right.$ at $\left.y^{2}=4 x\right)$

$y=m x-m^{2}\left(\operatorname{tangent}\right.$ at $\left.x^{2}=4 y\right)$

$\frac{1}{m}=-m^{2}$ (for common tangent)

$m^{3}=-1$

$\mathrm{m}=-1$

$y=-x-1$

$x+y+1=0$

This line touches circle

$\therefore$ apply $\mathrm{p}=\mathrm{r}$

$c=\left|\frac{0+0+1}{\sqrt{2}}\right|=\frac{1}{\sqrt{2}}$

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