1.

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

Answer»

`(0,1)`
`(2,OO)`
`(3,oo)`
`(0,oo)`

Solution :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`


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