The locus of the mid-points of the chords of the circle x^2+y^2=25 which are tangent to th…

Question

The locus of the mid-points of the chords of the circle $x^2+y^2=25$ which are tangent to the hyperbola $\dfrac{x^2}9-\dfrac{y^2}{16}=1$ is

Accepted Answer

Chord with midpoint $(h,k)$: $hx+ky=h^2+k^2$ (equation $T=S_1$). For this line ($lx+my=n$ form) to be tangent to $\dfrac{x^2}9-\dfrac{y^2}{16}=1$, the condition is $n^2=9l^2-16m^2$:
$(h^2+k^2)^2=9h^2-16k^2.$

$(x^2+y^2)^2-9x^2+16y^2=0.$

Solutions · 1