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The locus of middle points of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is
${({x^2} - {y^2})^2} = {a^2}{x^2} + {b^2}{y^2}$
${({x^2} + {y^2})^2} = {a^2}{x^2} + {b^2}{y^2}$
${({x^2} - {y^2})^2} = {a^2}{x^2} - {b^2}{y^2}$
${({x^2} + {y^2})^2} = {a^2}{x^2} - {b^2}{y^2}$
Solution
Equation of chord of the given circle whose middle point is
$(\mathrm{h}, \mathrm{k})$ is $\mathrm{xh}+\mathrm{yk}=\mathrm{h}^{2}+\mathrm{k}^{2}$
and equation of tangent line
upon hyperbola is $\mathrm{y}-\mathrm{mx}=\sqrt{\mathrm{a}^{2} \mathrm{m}^{2}-\mathrm{b}^{2}}$
As both lines are identical
$\Rightarrow-\frac{m}{h}=\frac{1}{k}=\frac{\sqrt{a^{2} m^{2}-b^{2}}}{h^{2}+k^{2}}$
Eliminating $m$
$\Rightarrow\left(\mathrm{h}^{2}+\mathrm{k}^{2}\right)^{2}=\left(\mathrm{a}^{2} \mathrm{h}^{2}-\mathrm{b}^{2} \mathrm{k}^{2}\right)$
hence the locus is $\left(x^{2}+y^{2}\right)^{2}=\left(a^{2} x^{2}-b^{2} y^{2}\right)$
