The locus of the midpoints of the chord of the circle, $$x^2 + y^2 = 25$$ which is tangent to the hyperbola, $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$ is:

Let the midpoint of a chord of the circle $$x^2 + y^2 = 25$$ be $$(h, k)$$. The equation of the chord with this midpoint is $$hx + ky = h^2 + k^2$$.

Rewriting this as a line in slope-intercept form: $$y = -\frac{h}{k}x + \frac{h^2 + k^2}{k}$$, where the slope is $$m = -\frac{h}{k}$$ and the y-intercept is $$c = \frac{h^2 + k^2}{k}$$.

For this line to be tangent to the hyperbola $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$, the tangency condition requires $$c^2 = 9m^2 - 16$$. Substituting the values of $$m$$ and $$c$$:

$$\frac{(h^2 + k^2)^2}{k^2} = 9 \cdot \frac{h^2}{k^2} - 16$$

Multiplying both sides by $$k^2$$: $$(h^2 + k^2)^2 = 9h^2 - 16k^2$$.

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