The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to th…

Question

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 :

Accepted Answer

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

Solutions · 1