The equations of the common tangents to the parabola `y = x^2 and y=-(x-2)^2` is/are :

Options

A. y=4(x-1)
B. y=0
C. y=-4(x-1)
D. y=-30x-50

Answer

A, B

Explanation

1,2
If y=mx+c is tangent to `y=x^(2)`, then `x^(2)-mx-c=0` has equal root. So,
`m^(2)+4c=0`
`orc=-(m^(2))/(4)`
So, the tangent to `y=x^(2)` is
`y=mx-(m^(2))/(4)`
Since this is also tangent to `y=-(x-2)^(2)`,
`mx-(m^(2))/(4)=-x^(2)+4x-4`
has equal roots. So,
`x^(2)+(m-4)x+(4-(m^(2))/(4))=0`
has equal roots. So,
`(m-4)^(2)-4(4-(m^(2))/(4))=0`
`orm^(2)=8m+16+m^(2)-16=0`
`orm=0,4`
So,, y=0 and y=4x-4 is the tangent.