The locus of a point such that two tangents d | vedclass

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(A) The equation of a tangent to the parabola $y^2 = 4ax$ with slope $m$ is $y = mx + \frac{a}{m}$.If this tangent passes through the point $(h, k)$,t

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$y^2 = \frac{9}{2}ax$

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(A) The equation of a tangent to the parabola $y^2 = 4ax$ with slope $m$ is $y = mx + \frac{a}{m}$.If this tangent passes through the point $(h, k)$,then $k = mh + \frac{a}{m}$,which simplifies to the quadratic equation $m^2h - km + a = 0$.Let the slopes of the two tangents be $m$ and $2m$.From the properties of quadratic equations,the sum of the roots is $m + 2m = 3m = \frac{k}{h}$ and the product of the roots is $m 	imes 2m = 2m^2 = \frac{a}{h}$.From the first equation,$m = \frac{k}{3h}$.Substituting this into the second equation: $2(\frac{k}{3h})^2 = \frac{a}{h}$.$2(\frac{k^2}{9h^2}) = \frac{a}{h}$.$2k^2 = 9ah$.Replacing $(h, k)$ with $(x, y)$,the locus is $2y^2 = 9ax$,or $y^2 = \frac{9}{2}ax$.

The locus of a point such that two tangents drawn from it to the parabola $y^2 = 4ax$ are such that the slope of one is double the other is:

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