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 $(a,b,c)$

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

Answer 1

Answer: (A), (B), (C), (D)

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)
Ans. (1), (2), (3), (4)
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 triad (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