If `y^2=4b(x-c) and y^2 =8ax` having common normal then `(a,b,c)` is

Options

A. `((1)/(2), 2,0)`
B. (1,1,0)
C. (1,1,3)
D. `((1)/(2),2,3)`

Answer

C

Explanation

Normal to parabola `y^(2) = 4 ax` is given by
`y = mx - 2 am - am^(3)`
`:.` Normal to parabola
`y^(2) = 4b (x - c)` is
`y = m(x - c) - 2bm - bm^(3)`
[replacing a by b and x by x - c]
`= mx - (2b + c) m = bm^(3)`

and normal ot parabola `y^(2) = 8` ax is
`y = mx - 4a - 2am^(3)`
[replacing a by 2a]
For common normal, we should have
`mx - 4am - 2am^(3) = mx - (2b + c) m - bm^(3)`
[using Eqs. (i) and (ii)]
`4 am + 3am^(2) = (2b + c) m + bm^(3)`
`implies (2a - b) m^(3) + (4a - 2b - c) m = 0`
`implies m((2a - b) m^(2) + (4a - 2b - c)) = 0`
`implies m = 0`
or `m^(2) = (2b + c+ 4a)/(2a - b) = (c )/(2a - b) - 2`
As, `m^(2) gt 0`, therefore `(C )/(2a - b) gt 2`
Note that if m = 0 then all options satisfy
(`:' y = 0` is common normal) and if common normal is other than the axis, then only option (c ) satisfies
[`:'` for option (c ), `2a - b = (3)/(2 - 1) = 3 gt 2`]