The locus of the midpoints of the parallel ch | vedclass

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(D) The equation of a chord of the hyperbola $xy = c^2$ with midpoint $(h, k)$ is given by $T = S_1$,which is $xh + yk = 2c^2$ is incorrect; the corre

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$my + x = 0$

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(D) The equation of a chord of the hyperbola $xy = c^2$ with midpoint $(h, k)$ is given by $T = S_1$,which is $xh + yk = 2c^2$ is incorrect; the correct formula for the chord with midpoint $(h, k)$ for $xy = c^2$ is $xh + yk = 2h k$.Comparing this to the slope-intercept form $y = mx + c'$,we have $y = -(\frac{h}{k})x + 2h$.Given the gradient of the chord is $m$,we have $m = -\frac{h}{k}$.Thus,$h = -mk$.Replacing $(h, k)$ with the general point $(x, y)$,we get $x = -my$,which implies $x + my = 0$.

The locus of the midpoints of the parallel chords with gradient $m$ of the rectangular hyperbola $xy = c^2$ is

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