If t_1 and t_2 are the parameters of the end | vedclass

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(C) For a parabola $y^2 = 4ax$, the coordinates of any point on it can be represented as $(at^2, 2at)$.Let the two end points of a focal chord be $P(a

Vedclass · Accepted Answer

$t_1 t_2 = -1$

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(C) For a parabola $y^2 = 4ax$, the coordinates of any point on it can be represented as $(at^2, 2at)$.Let the two end points of a focal chord be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$.The slope of the chord $PQ$ is $m = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2(t_2 - t_1)}{(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1 + t_2}$.Since the chord passes through the focus $(a, 0)$, the slope is also $m = \frac{2at_1 - 0}{at_1^2 - a} = \frac{2at_1}{a(t_1^2 - 1)} = \frac{2t_1}{t_1^2 - 1}$.Equating the slopes: $\frac{2}{t_1 + t_2} = \frac{2t_1}{t_1^2 - 1}$.$t_1^2 - 1 = t_1(t_1 + t_2) = t_1^2 + t_1 t_2$.$-1 = t_1 t_2$.Therefore, $t_1 t_2 = -1$.

If $t_1$ and $t_2$ are the parameters of the end points of a focal chord for the parabola $y^2 = 4ax$, then which one is true?

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