If y=mx+4 is a tangent to both the parabolas y^2=4x and x^2=2by, then b is equal to:

Question

If $y=mx+4$ is a tangent to both the parabolas $y^2=4x$ and $x^2=2by$, then $b$ is equal to:

Accepted Answer

Tangent to $y^2=4x$: $y=mx+\dfrac1m$; matching $c=4$: $m=\dfrac14$.

Line $y=\dfrac x4+4$ tangent to $x^2=2by$: substituting, $2x^2-bx-16b=0$; discriminant $=0$:
$$b^2+128b=0\Rightarrow b=-128\ (b=0\text{ rejected}).$$

Solutions · 1