Locus of mid points of chords of hyperbola x^ | vedclass

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Question

Let mid point be $(\mathrm{h}, \mathrm{k})$

chord of hyperbola: $\mathrm{hx}-\mathrm{ky}=\mathrm{h}^{2}-\mathrm{k}^{2}$

this is tangent to $\mat

Vedclass · Accepted Answer

independent of $a$ but dependent on $b.$

Vedclass · Answer

Let mid point be $(\mathrm{h}, \mathrm{k})$

chord of hyperbola: $\mathrm{hx}-\mathrm{ky}=\mathrm{h}^{2}-\mathrm{k}^{2}$

this is tangent to $\mathrm{x}^{2}=4 \mathrm{by}$

form of tangent for parabola: $\mathrm{y}=\mathrm{mx}-\mathrm{bm}^{2}$

comparing we get

$\mathrm{m}=\frac{\mathrm{h}}{\mathrm{k}} ;-\mathrm{bm}^{2}=-\left(\frac{\mathrm{h}^{2}-\mathrm{k}^{2}}{\mathrm{k}}ight)$

$	herefore \mathrm{b}\left(\frac{\mathrm{h}^{2}}{\mathrm{k}^{2}}ight)=-\left(\frac{\mathrm{h}^{2}-\mathrm{k}^{2}}{\mathrm{k}}ight)$

clearly, locus is dependent on $b,$ but not $a$

Locus of mid points of chords of hyperbola $x^2 -y^2 = a^2$ which are tangents to the parabola $x^2 = 4by$ will be -

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