"The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :"

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"tangent of hyperbola

$y = mx \pm \sqrt {9{m^2} - 16}$ ..... (i)

which is a chord of circle with mid-point (h, k)

so equation of chord T = S₁

hx + ky = h² + k²

$y = - {{hx} \over k} + {{{h^2} + {k^2}} \over k}$ ..... (ii)

by (i) and (ii)

$m = - {h \over k}$ and $\sqrt {9{m^2} - 16}$ = ${{{h^2} + {k^2}} \over k}$

$$9{{{h^2}} \over {{k^2}}} - 16 = {{{{\left( {{h^2} + {k^2}} \right)}^2}} \over {{k^2}}}$$

$\therefore$ Locus : 9x² $-$ 16y² = (x² + y²)²"

The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :

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The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :

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The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :