Question

Find the length of the perpendicular drawn from the origin on the tangent to the curve $x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$ at $\theta =\frac{\pi }{2}$.

Answer 1

Given curve is $x=a(\theta -\sin \theta ),y=a(1-\cos \theta )$

At $\theta =\frac{\pi }{2},x=a(\pi /2-\sin \pi /2)=\frac{a}{2}(\pi -2),y=a(1-\cos \pi /2)=a$

$\frac{dx}{d\theta }=a(1-\cos \theta ),\frac{dy}{d\theta }=a\sin \theta$

$\frac{dy}{dx}{|}_{x=\pi /2}=\frac{\frac{dy}{d\theta }{|}_{\pi /2}}{\frac{dx}{d\theta }{|}_{\pi /2}}=\frac{a(1-\cos \pi /2)}{a\sin \pi /2}=1$

Equation of the tangent at $\theta =\frac{\pi }{2}$ is $y-a=1(x-\frac{a}{2}(\pi -2\left)\right)$

$⟹2x-2y-a(\pi -4)=0$

As we know that

Distance point $(x_1,y_1)$ from line $ax+by+c=0$ is $\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$

Length of perpendicular drawn from the origin will be the distance of the line from origin

$⟹$ Length of the perpendicular is $\frac{a(4-\pi )}{\sqrt{2^2+(-2{)}^2}}=\frac{\sqrt{2}a}{4}(\pi -4)$