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

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(D) Let the midpoint of the chord be $(h, k).$The equation of the chord of the hyperbola $x^2 - y^2 = a^2$ with midpoint $(h, k)$ is given by $T = S_1

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

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(D) Let the midpoint of the chord be $(h, k).$The equation of the chord of the hyperbola $x^2 - y^2 = a^2$ with midpoint $(h, k)$ is given by $T = S_1$,which is $hx - ky = h^2 - k^2.$Rearranging this into the slope-intercept form $y = mx + c$,we get $ky = hx - (h^2 - k^2)$,or $y = \frac{h}{k}x - \frac{h^2 - k^2}{k}.$This line is a tangent to the parabola $x^2 = 4by$,which can be written as $y = \frac{1}{4b}x^2.$ The condition for the line $y = mx + c$ to be a tangent to the parabola $x^2 = 4by$ is $c = -bm^2.$Comparing the line $y = \frac{h}{k}x - \frac{h^2 - k^2}{k}$ with $y = mx + c$,we have $m = \frac{h}{k}$ and $c = -\frac{h^2 - k^2}{k}.$Substituting these into the condition $c = -bm^2$,we get $-\frac{h^2 - k^2}{k} = -b\left(\frac{h}{k}\right)^2.$Simplifying,we get $h^2 - k^2 = b\frac{h^2}{k},$ or $k(h^2 - k^2) = bh^2.$Replacing $(h, k)$ with $(x, y)$,the locus is $y(x^2 - y^2) = bx^2.$Clearly,this locus depends on $b$ but is independent of $a.$

The locus of the midpoints of the chords of the hyperbola $x^2 - y^2 = a^2$ which are tangents to the parabola $x^2 = 4by$ will be -

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