If the common tangent to the parabolas y^2=4x and x^2=4y also touches the circle x^2+y^2=c…

Question

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:

Accepted Answer

Tangent to $y^2=4x$: $y=mx+\dfrac1m$. Tangent to $x^2=4y$: $y=mx-m^2$. Equating slopes' intercepts: $\dfrac1m=-m^2\Rightarrow m=-1$.

Common tangent: $x+y+1=0$. Distance from origin: $c=\dfrac1{\sqrt2}$.

Solutions · 1