Let a line L:2x+y=k, k0 be a tangent to the hyperbola x^2-y^2=3.

Question

Let a line $L:2x+y=k$, $k>0$ be a tangent to the hyperbola $x^2-y^2=3$. If $L$ is also a tangent to the parabola $y^2=\alpha x$, then $\alpha$ is equal to

Accepted Answer

Tangency to $x^2-y^2=3$: $y=-2x+k$, condition $k^2=3(4)-3=9\Rightarrow k=3$. Line: $2x+y=3$.

Tangency to $y^2=\alpha x$ ($4A=\alpha$): condition $c=\dfrac A m$ with $m=-2,c=3$: $3=\dfrac{\alpha/4}{-2}\Rightarrow\alpha=-24$.

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