Question

Tangents are drawn from any point on the hyperbola `(x^2)/9-(y^2)/4=1` to the circle `x^2+y^2=9` . Find the locus of the midpoint of the chord of contact.

Answer 1

Let the midpoint of chord of contact is`(h,k)`
Parametric point on given hyperbola is`P(3sectheta, 2tantheta)`
Equation of tangent to given circle from the point P is,
`T=0`
`3xsectheta +2ytantheta=9 => (1)` Equation of tangent with the mid point(h,k) to the circle `x^2+y^2=9` is
`T=S1`
`hx+ky-9=h^2+k^2-9`
`hx+ky=h^2+k^2 => (2)`
Equations (1) And (2) are similar, So
`3sectheta/h=2tantheta/k=9/(h^2+k^2)` `sectheta=3h/(h^2+k^2)` and `tantheta=4.5k/(h^2+k^2)`
Putting these in,`sectheta^2-tantheta^2=1`
We get, `36h^2-81k^2=4((h^2+k^2)^2)` Hence, The locus is:=`x^2/9-y^2/4=((x^2+y^2)^2)/81`