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)

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