Thank you for reporting, we will resolve it shortly
Q.
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 $(a,b,c)$
Normal to these two curves are
$y = m(x - c) - 2bm - bm^3,$
$y = mx - 4am - 2am^3$
If they have a common normal
$(c + 2b) m + bm^3 = 4am + 2am^3$
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