Question

If the chords of contact of tangents from two poinst `(x_1, y_1)` and `(x_2, y_2)` to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` are at right angles, then find the value of `(x_1x_2)/(y_1y_2)dot`

Answer 1

Correct Answer - `-a^(4)//b^(4)`
The equations of the chords of contact of tangents drawn from `(x_(1),y_(1)) and (x_(2),y_(2))` to the ellipse
`(x^(2))/(b^(2))+(y^(2))/(b^(2))=1`
are `(x x_(1))/(a^(2))+(yy_(1))/(b^(2))=1" "(1)`
`(x x_(2))/(a^(2))+(yy_(2))/(b^(2))=1" "(2)`
It is given that (1) and (2) are at right angles. Therefore, `(-b^(2))/(a^(2))(x_(1))/(y_(1))xx(-b^(2))/(a^(2))(y_(2))/(y_(2))=-1`
or `(x_(1)y_(1))/(y_(1)y_(2))=-(a^(2))/(b^(4))`