Question

The angle at which the curve $y=k{e}^{kx}$ intersects the $y$ -axis is

• A. ${\tan }^{-1}\left(k^2\right)$
• B. ${\cot }^{-1}\left(k^2\right)$
• C. ${\sin }^{-1}\left(1/\sqrt{1+k^4}\right)$
• D. ${\sec }^{-1}\left(1/\sqrt{1+k^4}\right)$

Answer 1

The correct option is B ${\cot }^{-1}\left(k^2\right)$

$\frac{dy}{dx}=k^2{e}^{kx}$.

The curve intersects $y$-axis at $(0,k)$

So, $\frac{dy}{dx}{|}_{(0,k)}=k^2$.

If $\theta$ is the angle at which the given

curve intersects the $y$-axis then $\tan (\pi /2-\theta )=\frac{k^2-0}{1+0.k^2}=k^2$.

Hence $\theta ={\cot }^{-1}{k}^2$