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

The correct option is B 2

$y=x^2;y=-x^2+4x-4$


Let the commom tangent be,
$y=mx+c$
Finding the intersection with the parabola $y=x^2$
$x^2=mx+c\Rightarrow x^2-mx-c=0$
Condition for tangency,
$D=0\Rightarrow m^2+4c=0\cdots \left(1\right)$
Finding the intersection with the parabola $y=-x^2+4x-4$,
$-x^2+4x-4=mx+c\Rightarrow x^2+x(m-4)+(4+c)=0$
Condition for tangency,
$D=0\Rightarrow (m-4{)}^2-4(4+c)=0\Rightarrow m^2-8m+16-16-4c=0$
Using (1),
$\Rightarrow 2m^2-8m=0\Rightarrow m=0,4$

Hence, there are two common tangents to both the curves.