Question

The tangent to the curve given by $x=e^{\theta} \cdot \cos \theta, y=e^{\theta} \cdot \sin \theta$ at $\theta=\frac{\pi}{4}$ makes an angle with $X-$ axis is _________.

• A. $\frac{\pi}{2}$
• B. $0$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (A)

$x=e^{\theta} \cos \theta $
$\Rightarrow \frac{d x}{d \theta}=e^{\theta} \cos \theta-e^{\theta} \sin \theta $
$=e^{\theta}(\cos \theta-\sin \theta) $
$y=e^{\theta} \sin \theta $
$\Rightarrow \frac{d y}{d \theta}=e^{\theta} \sin \theta+e^{\theta} \cos \theta$
$=e^{\theta}(\sin \theta+\cos \theta)$
$=\frac{d y}{d x}=\frac{\sin \theta+\cos \theta}{\cos \theta-\sin \theta}$
Not defined at $\theta=\frac{\pi}{4}$
$\therefore \theta=\frac{\pi}{2}$ (Angle formed by tangent with $X$-axis)