given , `x^(2)+ 4y^(2)=4 or (x^(2))/(4)+(y^(2))/(1)=1`
Equation of any tangent to the ellipse on(i) can be written as
`(x)/(2) cos theta + y sin theta =1`
Equation of second ellipse is
`x^(2)+2y^(2)=6`
`implies (x^(2))/(6)+(y^(2))/(3)=1`
Suppose the tangent at P and Q meets at (h,k) Equation of the chord of contact of the tangents through A (h,k) is
`(hx)/(6)+(ky)/(3)=1`
But Eqs . (iv) and (ii) represent the same straight line , so comparing Eqs. (iv) adn (ii) we get
`(h//6)/( cos theta//2)=(k//3)/( sin theta)=(1)/(1)`
`implies h= 3 cos theta and k=3 sin theta`
therefore , coordinates of A are ( 3 cos,`theta,3 sin theta)`
Now , the joint equation of the tangents At A is given by `T^(2)=SS_(1)`,
`i.e., ((hx)/(6)+(ky)/(3)-1)^(2)=((x^(2))/(6)+(y^(2))/(3)-1)((h^(2))/(6)+(h^(2))/(3)-1)`
in Eq. (v) Coefficient of `x^(2)=(h^(2))/(36)-(1)/(6)((h^(2))/(6)+(h^(2))/(3)-1))`
`=(h^(2))/(36)-(h^(2))/(36)-(k^(2))/(18)+(1)/(6)=(1)/(6)-(k^(2))/(18)`
and coefficient of `y^(2)=(k^(2))/(9)-(1)/(3)((h^(2))/(6)+(k^(2))/(3)-1)`
`=(k^(2))/(9)-(h^(2))/(18)-(k^(2))/(9)+(1)/(3)=-(h^(2))/(18+(1)/(3)`
Again , coefficient of `x^(2)+` coefficient of `y^(2)`
`=-(1)/(18)(h^(2)+k^(2))+(1)/(6)+(1)/(3)`
`=-(1)/(18)( 9 cos ^(2) theta + 9 sin ^(2)theta+(1)/(2)`
`=-(9)/(18)+(1)/(2)=0`
which shows that two lines represent by Eq. (v) are at right angles to each other .
