Question

For the equation $x^{2/3}+y^{2/3}=a^{2/3}$, find the equation of tangent at the point $x=a{\sin }^3\theta ,y=a{\cos }^3\theta$.

• A. $y-a{\cos }^3\theta =\frac{\cos \theta }{\sin \theta }(x-a{\sin }^3\theta )$
• B. $y-a{\cos }^3\theta =-\frac{\cos \theta }{\sin \theta }(x-a{\sin }^3\theta )$
• C. $y-a{\cos }^3\theta =-\frac{\cos \theta }{\sin \theta }(x+a{\sin }^3\theta )$
• D. none of these

Answer 1

The correct option is B $y-a{\cos }^3\theta =-\frac{\cos \theta }{\sin \theta }(x-a{\sin }^3\theta )$

$x^{2/3}+y^{2/3}=a^{2/3}$

Differentiating with respect to x, gives us

$\frac{2}{3}[x^{-1/3}+y^{-1/3}{y}^{\prime }]=0$

Therefore

$y^{\prime }=-\frac{x^{-1/3}}{y^{-1/3}}$

$=-{\left(\frac{y}{x}\right)}^{1/3}$

At $x=asi{n}^3\theta$ and $y=aco{s}^3\theta$

$y^{\prime }=-\frac{cos\theta }{sin\theta }$

This is the slope of the tangent at the given point.

Hence the equation of the tangent is

$(y-aco{s}^3\theta )=\frac{-cos\theta }{sin\theta }(x-asi{n}^3\theta )$ [slope-point form]