Parabola Part 1

Normal to these two curves are
y = m(x – c) – 2bm – bm3,
y = mx – 4am – 2am3
If they have a common normal
(c + 2b) m + bm3 = 4am + 2am3
Now (4a – c – 2b) m = (b – 2a) m3
We get all options are correct for m = 0
(common normal x-axis)
Ans. (1), (2), (3), (4)
Remark :
If we consider question as
If the parabolas y2 = 4b(x – c) and y2 = 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 ≠ 0 : (4a - c - 2b) = (b - 2a) m2
m2 =

c

2a−b

- 2 > 0 ⇒

c

2a−b

> 2
Now according to options , option 4 is correct