The locus of the mid point of chords of the circle $x^{2}+y^{2}=a^{2}$ , which are tangent to the hyperbola $\frac {x^{2}}{a^{2}}-\frac {y^{2}}{b^{2}} = 1$ is

Let the midpoint of a chord of the circle be (h, k). Using the equation of the chord whose midpoint is (h, k), we get hx + ky = h squared plus k squared. If this chord is tangent to the hyperbola x squared over a squared minus y squared over b squared equals 1, applying the condition for tangency yields (h squared minus a squared) over a squared plus (k squared plus b squared) over b squared equals (h squared plus k squared) squared over (a squared times b squared). Simplifying this relationship gives h squared plus k squared equals a squared minus b squared, meaning the locus is x squared plus y squared equals a squared minus b squared.

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