Question

The equation of common tangent(s) to the parabola $y=x^2$ and $y=-{(x-2)}^2$ is/are

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

Answer 1

Answer: (A), (B)

The correct options are
A $y=4(x-1)$
B $y=0$

The equation of a tangent to the parabola $x^2=4ay$ can be written as $y=mx-a{m}^2$
For the parabola $y=x^2$, we can write the equation of the tangent as -
$y=mx-\frac{m^2}{4}$
Similarly, for the parabola $y=-(x-2{)}^2$, the equation of the tangent can be written as -
$y=m(x-2)+\frac{m^2}{4}$
$y=mx+(\frac{m^2}{4}-2m)$
As the tangent is common to both the parabola , both the equations should represent the same line.
Hence,
$-\frac{m^2}{4}=(\frac{m^2}{4}-2m)$
$m(m-4)=0$
$m=0$ or $m=4$
The common tangents are $y=0$ and $y=4x-4$