The equation of normal to the curve `x^(2//3)+y^(2//3)=a^(2//3)` at `(asin^(3) theta, a cos^(3) theta)` is

Options

A. `sin theta x-cos theta y=a sin^(4) theta-cos^(4)theta`
B. `sin thetax+cos thetay=a sin^(4) theta+a cos^(4)theta`
C. `sin thetax-cos thetay=asin^(4) theta+a cos^(4) theta`
D. None of the above

Answer

A

Explanation

Given curve is `x^(2/3)+y^(2/3)=a^(2/3)`
`:.2/3x^((-1)/3)+2/3y^((-1)/3)(dy)/(dx)=0implies(dy)/(dx)=(-y^(1/3))/(x^(1/3))`
At `(a sin^(3) theta, a cos^(3)theta)`,
`(dy)/(dx)=-(cos theta)/(sin theta)=-cot theta`
`:.` The slope of the normal is `tan theta`.
`:.` The equation of the normal is
`y-a cos^(3) theta=tan theta(x-asin^(3) theta)`
`impliescos thetay-a cos^(4) theta=sin theta x-a sin^(4)theta`
`implies sin theta x -cos theta y =a sin^(4) theta-a cos^(4)theta`.