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

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Question

(C) Let the midpoint of the chord be $(h, k)$. The equation of the chord of the hyperbola $x^{2}-y^{2}=4$ with midpoint $(h, k)$ is given by $T=S_{1}$

Vedclass · Accepted Answer

$y^{2}(x-2)=x^{3}$

Vedclass · Answer

(C) Let the midpoint of the chord be $(h, k)$. The equation of the chord of the hyperbola $x^{2}-y^{2}=4$ with midpoint $(h, k)$ is given by $T=S_{1}$,which is $xh-yk=h^{2}-k^{2}$.Rewriting this in the slope-intercept form $y=mx+c$,we get $y=\frac{h}{k}x - \frac{h^{2}-k^{2}}{k}$.This line touches the parabola $y^{2}=8x$ (where $a=2$). The condition for the line $y=mx+c$ to touch the parabola $y^{2}=4ax$ is $c=\frac{a}{m}$.Substituting $m=\frac{h}{k}$ and $c=-\frac{h^{2}-k^{2}}{k}$,we get $-\frac{h^{2}-k^{2}}{k} = \frac{2}{h/k} = \frac{2k}{h}$.Simplifying,$-(h^{2}-k^{2})h = 2k^{2}$,which gives $h^{2}h - k^{2}h = -2k^{2}$,or $h^{3} = k^{2}(h-2)$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^{2}(x-2)=x^{3}$.

The locus of the midpoints of the chords of the hyperbola $x^{2}-y^{2}=4$,which touch the parabola $y^{2}=8x$,is:

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