Question

If $y=x+4$ is a common tangent to the curves $y=e^x+3$ and $y=-a{x}^2,$ then the value of $a$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{16}$

Answer 1

The correct option is D $\frac{1}{16}$

If $y=x+4$ is a common tangent to the curves $y=e^x+3$ and $y=-a{x}^2$
We can see that $y=x+4$ is tangent to $y=e^x+3$ at $(0,4)$
Now, for tangent on $y=-a{x}^2$
$\Rightarrow x+4=-a{x}^2$
$\Rightarrow a{x}^2+x+4=0$
$\Rightarrow 1-16a=0(\because D=0)$
$\therefore a=\frac{1}{16}$