Question

Prove that all the normals to the curve $x=a\cos t+at\sin t$ and $y=a\sin t-at\cos t$ are at a distance ${}^{\prime }{a}^{\prime }$ from the origin.($a\in R)$

Answer 1

Given, $x=a\cos t+at\sin t$.......(1) and $y=a\sin t-at\cos t$........(2)

or, $\frac{dx}{dt}=a\left(tcost\right)$ and $\frac{dy}{dt}=a\left(t\sin t\right)$

$\therefore \frac{dy}{dt}=at\tan t$

Slope of tangent$\frac{dy}{dx}=tant$

Now equation of normal to the curve represented by the equation (1) and (2) at the point $\left(x\right(t),y(t\left)\right)$ is

$(Y-y(t\left)\right)=-{\frac{dx}{dy}|}_{\left(x\right(t),y(t\left)\right)}(X-x(t\left)\right)$

or, $Y-y=-\cot t(X-x)$

or, $Y+X\cot t=y+x\cot t$

or, $Y+X\cot t=a(\sin t-t\cos t)+a\cot t(\cos t+t\sin t)$

or, $Y\cos t+X\sin t=a$........(3)

Now the distance of (3) from the origin is a