At a point P on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1` tangents PQ is drawn. If the point Q be at a distance `(1)/(p)` from the point P, where 'p' is distance of the tangent from the origin, then the locus of the point Q is

Options

A. (a) `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1+(1)/(a^(2)b^(2))`
B. (b) `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1-(1)/(a^(2)b^(2))`
C. (c) `(x^(2))/(a^(2))+(y^(2))/(b^(2))=(1)/(a^(2)b^(2))`
D. (d) `(x^(2))/(a^(2))-(y^(2))/(b^(2))=(1)/(a^(2)b^(2))`

Answer

A

Explanation

Equation of the tangent at P is
`(x -a cos theta)/(asin theta) = (y-b sin theta)/(-b cos theta)`

The distance of the tangent from the origin is
`p = |(ab)/(sqrt(b^(2)cos^(2)theta+a^(2)sin^(2)theta))|`
`rArr (1)/(p) = (sqrt(b^(2)cos^(2)theta+a^(2)sin^(2)theta))/(ab)`
Now the coordinates of the point Q are given as follows
`((x-a cos theta)/(-a sin theta))/(sqrt(b^(2)cos^(2)theta+a^(2)sin^(2)theta)) =((y-b sin theta)/(bcos theta))/(sqrt(b^(2)cos^(2)theta+a^(2)sin^(2)theta)) =(1)/(p) = (sqrt(b^(2)cos^(2)theta+a^(2)sin^(2)theta))/(ab)`
`rArr x = a cos theta -(a sin theta)/(ab)` and `y = b sin theta (b cos theta)/(ab)`
`rArr ((x)/(a))^(2) + ((y)/(b))^(2) =1+ (1)/(a^(2)b^(2))` is the required locus.