1.

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

Answer»

<P>`(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1+(1)/(a^(2)b^(2))`
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1-(1)/(a^(2)b^(2))`
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=(1)/(a^(2)b^(2))`
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=(1)/(a^(2)b^(2))`

Solution :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.


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