# If the common tangent to the parabolas, $y^{2}=4x$ and $x^{2}=4y$ also touches the circle, $x^{2}+y^{2}=c^{2},$ then $c$ is equal to : Option: 1   Option: 2 Option: 3 Option: 4

\begin{aligned} &\mathrm{y}=\mathrm{mx}+\frac{1}{\mathrm{~m}}\left(\text { tangent at } \mathrm{y}^{2}=4 \mathrm{x}\right)\\ &y=m x-m^{2}\left(\text { tangent at } x^{2}=4 y\right)\\ &\frac{1}{\mathrm{~m}}=-\mathrm{m}^{2} \text { (for common tangent) }\\ &m^{3}=-1\\ &m=-1 \end{aligned}

\begin{aligned} &y=-x-1\\ &x+y+1=0\\ &\text { This line touches circle }\\ &\therefore \text { apply } \mathrm{p}=\mathrm{r}\\ &\mathrm{c}=\left|\frac{0+0+1}{\sqrt{2}}\right|=\frac{1}{\sqrt{2}} \end{aligned}

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