Question

Find the locus of the-mid points of the chords of the circle `x^2 + y^2=16`, which are tangent to the hyperbola `9x^2-16y^2= 144`

Answer 1

Equation of hyperbola is `(x^(2))/(16)-(y^(2))/(9)=1`
Let (h,k) be the midpoint of the chord of the circle `x^(2)+y^(2)=16.`
So, the equation of the chord will be
`hx+ky=h^(2)+k^(2)`
`rArr" "y=underset(m)ubrace((-h)/(k))x+underset(c)ubrace((h^(2)+k^(2))/(k))`
This will touch the hyperbola if
`c^(2)=a^(2)m^(3)-b^(2)`
`rArr" "((h^(2)+k^(2))/(k))=16((-h)/(k))^(2)-9`
`rArr" "(h^(2)+k^(2))^(2)=16h^(2)-9k^(2)`
Therefore, required locus is `(x^(2)+y^(2))=16x^(2)-9y^(2)`.