1. Let the midpoint of the chord be $(h,k)$.

   For the circle $x^2+y^2=25,$ the chord whose midpoint is $(h,k)$ has equation $xh+yk=h^2+k^2.$

   This is the standard chord-with-given-midpoint form for the circle $x^2+y^2=a^2$.

2. This chord is tangent to the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1.$

   So the line $xh+yk=h^2+k^2$ must be a tangent to the hyperbola.

3. Use the tangent condition for the hyperbola.

   For the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ a line $lx+my=n$ is tangent iff $n^2=a^2l^2-b^2m^2.$

   Here, $l=h,\quad m=k,\quad n=h^2+k^2,\quad a^2=9,\quad b^2=16.$

   Therefore, $(h^2+k^2)^2=9h^2-16k^2.$

4. Hence the locus of $(h,k)$ is $(h^2+k^2)^2-9h^2+16k^2=0.$

   Replacing $(h,k)$ by $(x,y)$, the locus is $(x^2+y^2)^2-9x^2+16y^2=0.$

5. Match with the options.

   This is exactly Option A: $(x^2+y^2)^2-9x^2+16y^2=0.$

Verification with stored answer

Stored correct answer: A

Our derived answer: A

So they agree.