Tangent are drawn from the point (3, 2) to the ellipse `x^2+4y^2=9` . Find the equation to their chord of contact and the middle point of this chord of contact.

Explanation

`x^(2)+4y^(2)=9`
The equatin of the chord of contact of the pair of tangents from (3,2) is `3x+8y=9 " "(1)`

This mut be the same as the chord whose middle point is (h,k)
`T=S_(1)`
`(hx)/(9)+(ky)/(9//4)=(h^(2))/(9)+(k^(2))/(9//4)`
or hx+4ky=`h^(2)+4k^(2)" "(2)`
Equation (1) and (2) represent the same straight lines.
Comparing the coefficients of (1) and (2) , we get
`(h)/(3)=(4k)/(8)=(h^(2)+4k^(2))/(9)`
`or 2h=3k and 3h=h^(2)+4k^(2)`
or `3h=h^(2)+4xx(4h^(2))/(9)`
or `(25h^(2))/(9)=3h`
or `h=(27)/(25) and k=(18)/(25)`