The equation of the straight line passing through the point `(acos^3theta,asin^3theta)` and perpendicular to the line `xsectheta+ycos e ctheta=a` is (A) `xcostheta-ysintheta=acos2theta` (B) `xcostheta+ysintheta=acos2theta` (C) `xsintheta+ycostheta=acos2theta` (D) none of these

Options

A. `x cos theta-ysin theta=acos theta`
B. `x cos theta-ysin theta=acos 2theta`
C. `x cos theta-ysin theta=-acos 2theta`
D. none of these

Answer

B

Explanation

The equation of a in line perpendicular to the line `xsectheta + ycosec theta=a` is
`xcosec theta -ysec theta + lambda =0,lambda` is a constant `" '…(i)`
This line passes through `(acos^(3)theta, a sin^(3) theta)`.
`therefore a cos^(3) theta cosec theta - a sin^(3) theta sec theta + lambda =0`
`implies lambda =a(sin^(3)theta sec theta -cos^(3) theta cosec theta)`
Putting the value of `lambda` in (i), we get
`xcosec theta-ysec theta+a (sin^(3)theta sec theta -cos^(3)theta cosec theta)=0`
`implies (x)/(sintheta)-(y)/(cos theta)+a((sin^(3)theta)/(cos theta)-(cos^(3)theta)/(sin theta))=0`
`implies xcos theta-ysin theta+a(sin^(4)theta-cos^(4)theta)=0`
`implies xcos theta-ysin theta=a cos 2theta`