JEE PYQ: Parabola Question 24
Question 24 - 2019 (10 Jan Shift 1)
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 $(a, b, c)$?
(1) $\left(\frac{1}{2}, 2, 3\right)$
(2) $(1, 1, 3)$
(3) $\left(\frac{1}{2}, 2, 0\right)$
(4) All of these
Show Answer
Answer: (4)
Solution
Normal to $y^2 = 8ax$: $y = mx - 4am - 2am^3$. Normal to $y^2 = 4b(x-c)$: $y = m(x-c) - 2bm - bm^3$. For common normal: $4a + 2am^3 = cm + 2bm + bm^3$ or $m = 0$. Since $x$-axis ($m = 0$) is always a common normal, all options are correct.
BETA
