`
If the parabolas $y^{2}=4 b(x-c)$ and $y^{2}=8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$
Correct Option: 1, 2, 3, 4
Normal to these two curves are
$y=m(x-c)-2 b m-b m^{3}$
$y=m x-4 a m-2 a m^{3}$
If they have a common normal
$(c+2 b) m+b m^{3}=4 a m+2 a m^{3}$
Now $(4 a-c-2 b) m=(b-2 a) m^{3}$
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 $y^{2}=4 b(x-c)$ and $y^{2}=8 a x$ 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 $m \neq 0:(4 a-c-2 b)=(b-2 a) m^{2}$
$m^{2}=\frac{c}{2 a-b}-2>0 \Rightarrow \frac{c}{2 a-b}>2$
Now according to options, option 4 is correct
.jpg)