Show that the normal at any point θ to the curve `x = a costheta + a theta sin theta, y = a sintheta – atheta costheta` is at a constant distance from the origin.

Options

A. it makes a constant angle with the X-axis
B. it passes through the origin
C. it is at a constant distance from the origin
D. Name of the above

Answer

C

Explanation

Given, `x = a ( cos theta + theta sin theta)`
and `y = a ( sin theta - theta cos theta ) `
` therefore " " (dx)/(d theta) = a (-sin theta + sin theta + theta cos theta) = a theta cos theta `
and `(dy)/( dtheta) = a ( cos theta - cos theta + theta sin theta)`
`" " (dy)/(dtheta) = a theta sin theta rArr (dy)/( d x ) = tan theta `
Thus, equation of normal is
`" " (y - a ( sin theta - theta cos theta) )/( x- a ( cos theta + theta sin theta )) = ( - cos theta )/( sin theta)`
`rArr - x cos theta + a theta sin theta cos theta + a cos ^(2) theta = y sin theta + theta a sin theta - a sin ^(2) theta `
`rArr x cos theta + y sin theta = a`
whose distance from origin is
`" " (|0 + 0 -a|)/( sqrt( cos ^(2) theta + sin ^(2) theta ) ) = a `