Question

The length of the segment of the tangent line to the curve $x=a{\cos }^{3}t,y=a{\sin }^{3}t$, at any point on the curve cut off by the coordinate axes is

• A. $4a$
• B. $a$
• C. $a^2$
• D. $2a$

Answer 1

The correct option is B $a$

$x=a{\cos }^3\theta ,y=a{\sin }^3\theta$

$y^{\prime }=\frac{dy}{dx}=\frac{a\times 3{\sin }^3\theta \times \cos \theta }{a\times 3{\cos }^3\theta \times -\sin \theta }=-\tan \theta$

Tangent at $\theta$ is

$y-a{\sin }^3\theta =-\tan \theta (x-a{\cos }^3\theta )$

$\Rightarrow x\tan \theta +y=a{\sin }^3\theta +a\tan \theta {\cos }^3\theta$

$=a{\sin }^3\theta +a\sin \theta .{\cos }^2\theta$

$=a\sin \theta$

So, $\frac{x\tan \theta }{a\sin \theta }+\frac{y}{a\sin \theta }=1\Rightarrow \frac{x}{a\cos \theta }+\frac{y}{a\sin \theta }=1$

So the length of segment $=\sqrt{{\left(a\cos \theta \right)}^2+{\left(a\sin \theta \right)}^2}$

$=a$