Question

The sum of the length of sub tangent of sub tangent and tangent to the curve
$x=c[2cos\theta -log(cosec\theta +cot\theta )],y=csin2\theta at\theta =\frac{\pi }{3}is$

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

Answer 1

Answer: (C)

$\frac{3c}{2}$

Given,

$x=c[2\cos \theta -\log (\csc \theta +\cot \theta \left)\right]$

$\Rightarrow \frac{dx}{d\theta }=c(-2\sin \theta +\csc \theta )$

$y=c\sin 2\theta$

$\frac{dy}{d\theta }=2c\cos 2\theta$

Also, $y_1=c\sin \frac{2\pi }{3}=\frac{c\sqrt{3}}{2}$

Slope of tangent at $\theta =\frac{\pi }{3}$ is

$m={\frac{dy}{dx}}_{\theta =\frac{\pi }{3}}=\sqrt{3}$

Length of tangent $=\left|\frac{y_1\sqrt{1+m^2}}{m}\right|=c$

Length of subtangent $=\left|\frac{y_1}{m}\right|=\frac{c}{2}$

Required sum $=c+\frac{c}{2}=\frac{3c}{2}$