Question

If $y=mx+c$ is a common tangent to the curves $y=x^2+8$ and $y=-x^2,$ then which of the following(s) is/are correct

• A. $m=4,c=4$
• B. $m=-4,c=-4$
• C. $m=-4,c=4$
• D. $m=16,c=4$

Answer 1

Answer: (A), (C)

$m=-4,c=4$

If $y=mx+c$ is tangent to $y=x^2+8$
$\therefore mx+c=x^2+8$
$\Rightarrow x^2-mx+8-c=0$
$\Rightarrow m^2-4(8-c)=0(\because D=0)$
$\Rightarrow m^2+4c-32=0\cdots \left(i\right)$

$y=mx+c$ is also tangent to $y=-x^2$
$\Rightarrow mx+c=-x^2$
$\Rightarrow x^2+mx+c=0$
$\Rightarrow m^2-4(1\cdot c)=0(\because D=0)$
$\Rightarrow m^2-4c=0\cdots \left(ii\right)$
Solving equations $\left(i\right)$ and $\left(ii\right),$ we get
$m=\pm 4,c=4$