If the parabolas y2 = 4b(x -- c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?
Solutions · 1
Official solution
Normal to the two given curves are
y = m(x -- c) -- 2bm -- bm3,
y = mx -- 4am -- 2am3
If they have a common normal, then
(c + 2b)m + bm3
= 4am + 2am3
\[ \Rightarrow \] (4a -- c -- 2b) m = (b -- 2a)m3
\[ \Rightarrow \] (4a -- c -- 2b) = (b -- 2a)m2
\[ \Rightarrow \] m2 = \[{c \over {2a - b}} - 2\] > 0
By checking all options we found (A) is right option.
Note : We get that all the options are correct for m = 0
i.e., when common normal is x-axis. But may be in question they want common
normal other than x -- axis, hence answer is (A).