The locus of the midpoints of the chords of t | vedclass

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

Vedclass · Accepted Answer

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

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(B) Let the midpoint of the chord be $(h, k)$.The equation of the chord of the circle $x^2+y^2=16$ with midpoint $(h, k)$ is given by $T=S_1$,which is $xh+yk = h^2+k^2$.This chord is a tangent to the hyperbola $9x^2-16y^2=144$,which can be written as $\frac{x^2}{16} - \frac{y^2}{9} = 1$.The condition for the line $lx+my=n$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $n^2 = a^2l^2 - b^2m^2$.Here,$l=h$,$m=k$,$n=h^2+k^2$,$a^2=16$,and $b^2=9$.Substituting these values,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 midpoints of the chords of the circle $x^2+y^2=16$,which are tangents to the hyperbola $9x^2-16y^2=144$,is

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