Question

If the curves $y=2e^x$ and $y=a{e}^{-x}$ intersect orthogonally, then $a=$?

Answer 1

Step 1: Find the slope of the curves.

Given: $y=2e^x$

Differentiating with respect to $x$

$\Rightarrow$ $\frac{dy}{dx}=2e^x$

Thus, slope $m_1=2e^x$.

Given: $y=a{e}^{-x}$

Differentiating with respect to $x$

$\Rightarrow$ $\frac{dy}{dx}=-a{e}^{-x}$

thus, slope $m_2=-a{e}^{-x}$

Step $2$: Compute the value of $a$ using condition of orthogonality

Given: Both curves intersect orthogonally.

Thus, $m_1.m_2=-1$

$\Rightarrow$ $\left(2e^x\right)\left(-a{e}^{-x}\right)=-1$

$\Rightarrow$ $-2a=-1$

$\Rightarrow$ $a=\frac{1}{2}$

Hence, the value of a is $\frac{1}{2}$.