The equation to the line touching both the parabolas `y^2 =4x` and `x^2=-32y` is

Options

A. `x+2y+4=0`
B. `2x+y-4=0`
C. `x-2y-4=0`
D. `x-2y+4=0`

Answer

A

Explanation

The equation of any tangent to the parabola `y^(2)=4x` is
`y=mx+(1)/(m)....(i)`
This touches the parabola `x^(2)=-32y`, therefore the equation
`x^(2)=-32(mx+(1)/(m))` has equal roots
`therefore (32m)^(2)=4((32)/(m))`
`=Rightarrow 8m^(3)=1 Rightarrow m=(1)/(2)`
On putting the value of m in Eq. (i), we get
x-2y+4=0