The equation of the line touching both the parabolas `y^(2)=4xandx^(2)=-32y` is ax+by+c=0. Then the value of a+b+c is ___________ .

Answer

3

Explanation

(3) The equation of tangent to parabola `y^(2)=4x` is
`y=mx+(1)/(m)`
Since (1) is also the tangent of `x^(2)=-32y`, we have
`x^(2)=-32(mx+(1)/(m))`
`orx^(2)+32mx+(32)/(m)=0`
The above equation must have equal roots.
Hence, its discriminant must be zero. Therefore,
`(32m)^(2)=4xx1xx(32)/(m)`
`i.e.," "m^(3)=(1)/(8)or=(1)/(2)`
From (1),
`y=(x)/(2)+2`
`orx-2y+4=0`