If the parabolas y²=4b(x−c) and y²=8ax have a common normal, other than the x-axis, 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) − bm − bm³,
y = mx − 2am − 2am³
If they have a common normal
(c + 2b)m + bm³ = 4am + 2am³
Now (4a − c − b)m = (b − 2a)m³
We get all options are correct for m = 0 (common normal x-axis)
If we consider the question as:
If the parabolas y²=4b(x−c) and y²=8ax have a common normal other than the x-axis, then which one of the following is a valid choice for the ordered triad (a, b, c)?
When m ≠ 0:
(4a − c − b) = (b − 2a)m²
m² = (4a − c − b)/(b − 2a) > 0 ⇒ (4a − c − b)/(b − 2a) > 0
Now according to the options, option 4 is correct.