If the parabolas y^2=4b(x-c) and y^2=8ax have a common normal, then which one of the follo…

Question

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a,b,c)$?

Accepted Answer

Normal to $y^2=8ax$: $y=mx-4am-2am^3$. Normal to $y^2=4b(x-c)$: $y=m(x-c)-2bm-bm^3$.

Equating (same slope $m\ne0$, matching intercepts):
$$m^2=\frac{4a-c-2b}{b-2a}.$$

Testing the options, only $(a,b,c)=(1,1,3)$ gives $m^2=\dfrac{4-3-2}{1-2}=1>0$ --- a valid nontrivial common normal.

Solutions · 1