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

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(C) Let the focal chord pass through the focus $S(a, 0)$ and connect points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ on the parabola $y^2 = 4ax$.The

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the directrix

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(C) Let the focal chord pass through the focus $S(a, 0)$ and connect points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ on the parabola $y^2 = 4ax$.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 this chord with respect to the parabola $y^2 = 4ax$.The equation of the polar of $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$,or $yy_1 - 2ax = 2ax_1$.Comparing this with the chord equation $y(t_1 + t_2) - 2x = 2at_1t_2$,we get:$\frac{t_1 + t_2}{y_1} = \frac{-2}{-2a} = \frac{2at_1t_2}{2ax_1}$From $\frac{-2}{-2a} = \frac{2at_1t_2}{2ax_1}$,we get $x_1 = at_1t_2$.Since $t_1t_2 = -1$,we have $x_1 = -a$.Thus,the locus of the pole 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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