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. ${\sin }^{-1}\left(\frac{1}{\sqrt{1+k^4}}\right)$
• D. ${\sec }^{-1}\sqrt{1+k^4}$

Answer 1

Answer: (B)

Given $y=k{e}^{kx}$. The curve
intersects the $y$ -axis at $(0,k)$
So, ${\left(\frac{dy}{dx}\right)}_{(0,k)}=k^2$
If $\theta$ is the angle at which 1 the given curve intersects the $y$-axis, then
$\tan \left(\frac{\pi }{2}-\theta \right)=\frac{k^2-0}{1+0.k^2}=k^2$
$\Rightarrow \theta =co{t}^{-1}(k^2)$