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.

Explanation

Let any point on the hyperbola is `(3 sec theta, 2 tan theta).`
` therefore ` Chord of contact of the circle ` x^(2)+y^(2)=9` with respect to the point `(3 sec theta, 2 tan theta)` is,
`(3sec theta)x+(2 tan theta) y=9 " …(i)" `
Let `(x_(1),y_(1))` be the mid-point of the chord of contact.
`rArr` Equation of chord in mid-point form is
`x x_(1)+y y_(1)=x_(1)^(2)+y_(1)^(2) " ...(ii)" `
Since, Eqs. (i) and (ii) are identically equal.
` therefore (3sec theta)/(x_(1))=(2 tan theta)/(y_(1))`
`=(9)/(x_(1)^(2)+y_(1)^(2))`
`rArr sec theta=(9x_(1))/(3(x_(1)^(2)+y_(1)^(2)))`
`and tan theta=(9y_(1))/(2(x_(1)^(2)+y_(1)^(2)))`
Thus, eliminating `'theta'` froom above equation, we get
`(81x_(1)^(2))/(9(x_(1)^(2)+y_(1)^(2))^(2))-(81y_(1)^(2))/(4(x_(1)^(2)+y_(1)^(2))^(2))=1" "[ because sec^(2)theta-tan^(2)theta=1]`
`therefore " Required locus "(x^(2))/(9)-(y^(2))/(4)=((x^(2)+y^(2))^(2))/(81)`.