The locus of the poles of focal chords of the | vedclass

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(C) Let the parabola be $y^2 = 4ax$. Let $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ be two points on the parabola such that $PQ$ is a focal chord passi

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the directrix

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(C) Let the parabola be $y^2 = 4ax$. Let $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ be two points on the parabola such that $PQ$ is a focal chord passing through the focus $S(a, 0)$.The equation of the chord $PQ$ is $y(t_1 + t_2) = 2x + 2at_1t_2$.Since it passes through $(a, 0)$,we have $0 = 2a + 2at_1t_2$,which implies $t_1t_2 = -1$.Let $(x_1, y_1)$ be the pole of the chord $PQ$ with respect to the parabola $y^2 = 4ax$.The equation of the polar of $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$,which can be rewritten as $yy_1 - 2ax = 2ax_1$.Comparing this with the equation of the chord $y(t_1 + t_2) - 2x = 2at_1t_2$,we get:$\frac{y_1}{t_1 + t_2} = \frac{-2a}{-2} = \frac{2ax_1}{2at_1t_2}$From $\frac{y_1}{t_1 + t_2} = a$,we have $y_1 = a(t_1 + t_2)$.From $a = \frac{x_1}{t_1t_2}$,we have $x_1 = at_1t_2$.Since $t_1t_2 = -1$,we get $x_1 = a(-1) = -a$.Thus,the locus of the pole $(x_1, y_1)$ is $x = -a$,which is the directrix of the parabola.

The locus of the poles of focal chords of the parabola $y^2 = 4ax$ is

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