If the parabols `y^(2) = 4kx (k gt 0)` and `y^(2) = 4 (x-1)` do not have a common normal other than the axis of parabola, then k `in`
(a) `(0,1)` (b) `(2,oo)` (c) `(3,oo)` (d) `(0,oo)`

Options

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

Answer

A, B, C

Explanation

If the parabolas have a common normal of slope `m(m ne 0)` then it is given by
`y = mx - 2km -km^(3)`
and `y = m(x-1) -2m -m^(3) = mx -3m -m^(3)`
`rArr 2km + km^(3) = 3m + m^(3)`
`rArr m = 0, m^(2) =(3-2k)/(k-1)`.
If `m^(2) lt 0` then the only common normal is the axis.
`rArr (3-2k)/(k-1) lt 0`
`rArr (k-1) (2k-3) gt 0`
`k gt (3)/(2)` or `k lt 1` and `k gt 0`