Question

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

• A. $(1,1,0)$
• B. $(\frac{1}{2},2,3)$
• C. $(\frac{1}{2},2,0)$
• D. $(1,1,3)$

Answer 1

The correct option is D $(1,1,3)$

Normal to these two curves are
$y=m(x-c)-2bm-b{m}^3$,
$y=mx-4am-2a{m}^3$

If they have a common normal
$(c+2b)m+b{m}^3=4am+2a{m}^3$
Now $(4a-c-2b)m=(b-2a)m^3$

We get all options are correct for $m=0$
(common normal x-axis)

If we consider question as

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal other than x-axis, then which one of the following is a valid choice for the ordered traid (a, b, c)?
When $m\ne 0$: $(4a-c-2b)=(b-2a)m^2$
$m^2=\frac{c}{2a-b}-2>0\Rightarrow \frac{c}{2a-b}>2$

Now according to options, option $4$ is correct.