If (at^2, 2at) are the coordinates of one end | vedclass

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(D) For the parabola $y^2 = 4ax$,the coordinates of any point on it are given by $(at^2, 2at)$.If $t_1$ and $t_2$ are the parameters of the endpoints

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$\left( \frac{a}{t^2}, -\frac{2a}{t} \right)$

Vedclass · Answer

(D) For the parabola $y^2 = 4ax$,the coordinates of any point on it are given by $(at^2, 2at)$.If $t_1$ and $t_2$ are the parameters of the endpoints of a focal chord,then the condition is $t_1 t_2 = -1$.Given $t_1 = t$,the parameter of the other endpoint is $t_2 = -\frac{1}{t}$.Substituting $t_2 = -\frac{1}{t}$ into the parametric coordinates $(at_2^2, 2at_2)$,we get:$x = a\left(-\frac{1}{t}\right)^2 = \frac{a}{t^2}$$y = 2a\left(-\frac{1}{t}\right) = -\frac{2a}{t}$Thus,the coordinates of the other end are $\left( \frac{a}{t^2}, -\frac{2a}{t} \right)$.

If $(at^2, 2at)$ are the coordinates of one end of a focal chord of the parabola $y^2 = 4ax$,find the coordinates of the other end.

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