Question

The angle at which the curve $y = ke ^{ kx }$ 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 $y = ke ^{ kx }$. 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
$\tan\left(\frac{\pi}{2} -\theta \right) = \frac{k^2 - 0}{1 + 0.k^2} = k^2$
$\Rightarrow \theta = cot^{-1} (k^2)$