The locus of the mid-points of the chords of | vedclass

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(C) Let the mid-point of the chord be $(h, k)$. The equation of the chord of the circle $x^2+y^2=16$ with mid-point $(h, k)$ is given by $T=S_1$,which

Vedclass · Accepted Answer

$16x^2-9y^2=(x^2+y^2)^2$

Vedclass · Answer

(C) Let the mid-point of the chord be $(h, k)$. The equation of the chord of the circle $x^2+y^2=16$ with mid-point $(h, k)$ is given by $T=S_1$,which is $hx+ky=h^2+k^2$. Rearranging this,we get $y = -\frac{h}{k}x + \frac{h^2+k^2}{k}$. For the hyperbola $9x^2-16y^2=144$,we can write it as $\frac{x^2}{16} - \frac{y^2}{9} = 1$. Here $a^2=16$ and $b^2=9$. The condition for the line $y=mx+c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $c^2 = a^2m^2 - b^2$. Substituting $m = -\frac{h}{k}$ and $c = \frac{h^2+k^2}{k}$,we get: $(\frac{h^2+k^2}{k})^2 = 16(-\frac{h}{k})^2 - 9$. Multiplying by $k^2$,we get $(h^2+k^2)^2 = 16h^2 - 9k^2$. Replacing $(h, k)$ with $(x, y)$,the locus is $16x^2-9y^2=(x^2+y^2)^2$.

The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are the tangents to the hyperbola $9x^2-16y^2=144$ is

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