Question

Number of common tangents of $y = x^2$ and $y = -x^2 + 4x - 4$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

We have, equation of parabola $y=x^{2}$
Let $P\left(\alpha, \alpha^{2}\right)$ is a point on the parabola,
$\therefore y-\alpha^{2}=2 \alpha(x-\alpha)$
$\because \left[\because \frac{d y}{d x}=2 x \Rightarrow \frac{d y}{d x_{\left(\alpha, a^{2}\right)}}=2 \alpha\right]$
$\Rightarrow y=2 \alpha x-\alpha^{2}$
Also, given $y=-x^{2}+4 x-4$
$\therefore -x^{2}+4 x-4=20 x-\alpha^{2} $
$ \Rightarrow x^{2}+2 x(\alpha-2)+\left(4-\alpha^{2}\right)=0 $
Discriminant $=0 $
$ 4(\alpha-2)^{2}-4\left(4-\alpha^{2}\right)=0 $
$ \Rightarrow (\alpha-2)^{2}-\left(4-\alpha^{2}\right)=0 $
$\Rightarrow \alpha^{2}-4 \alpha+4-4+\alpha^{2}=0$
$ \Rightarrow \alpha^{2}-2 \alpha=0 $
$ \Rightarrow \alpha=0, \alpha=2$