Question

The angle formed by the abscissa and the tangent to the parabola $y=x^2+4x-17$ at the point $\left(\frac{5}{2},\frac{3}{4}\right)$ is

• A. ${\tan }^{-1}\left(4\right)$
• B. ${\tan }^{-1}\left(6\right)$
• C. ${\tan }^{-1}\left(9\right)$
• D. None of these.

Answer 1

The correct option is C

${\tan }^{-1}\left(9\right)$

Explanation for the correct option:

Find the angle:

Given the equation of the parabola $y=x^2+4x-17$

Differentiating w.r.t '$x$', we get

$$\begin{gathered} \frac{dy}{dx}=2x+4 \\ \Rightarrow m=2x+4\left[\because \frac{dy}{dx}=\text{slope}\right] \\ \Rightarrow \tan \left(\theta \right)=2x+4 \end{gathered}$$

Substitute $\left(\frac{5}{2},\frac{3}{4}\right)$

$$\begin{gathered} \tan \theta =2\left(\frac{5}{2}\right)+4 \\ \Rightarrow \tan \theta =9 \\ \Rightarrow \theta ={\tan }^{-1}\left(9\right) \end{gathered}$$

Hence, option (C) is the correct answer.