Question

The sum of the lengths of subtangent and tangent to the curve
$x=c[2\cos \theta -\log (\cos ec\theta +\cot \theta \left)\right]y=c\sin 2\theta$ at $\theta =\frac{\pi }{3}$ is

• A. $\frac{c}{2}$
• B. $2c$
• C. $\frac{3c}{2}$
• D. $\frac{5c}{2}$

Answer 1

The correct option is C $\frac{3c}{2}$

$x=c[2\cos \theta -\log (\cos ec\theta +\cot \theta \left)\right]$
$\Rightarrow \frac{dx}{d\theta }=c(-2sin\theta +cosec\theta )$
$y=c\sin 2\theta$
$\Rightarrow \frac{dy}{d\theta }=2c\cos 2\theta$
Also, $y_1=c\sin \frac{2\pi }{3}=\frac{c\sqrt{3}}{2}$
Slope of tangent at $\theta =\frac{\pi }{3}$ is
$m={\left(\frac{dy}{dx}\right)}_{\theta =\frac{\pi }{3}}=\sqrt{3}$
Length of tangent $=|\frac{y_1\sqrt{1+m^2}}{m}|=c$
Length of sub-tangent $=|\frac{y_1}{m}|=\frac{c}{2}$
Required Sum $=c+\frac{c}{2}=\frac{3c}{2}$