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

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(D) Let the parabola be $y^2 = 4ax$. Let the focal chord be a line passing through the focus $(a, 0)$. The equation of a chord with pole $(x_1, y_1)$

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The directrix

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(D) Let the parabola be $y^2 = 4ax$. Let the focal chord be a line passing through the focus $(a, 0)$. The equation of a chord with pole $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$. Since this chord passes through the focus $(a, 0)$,we substitute $x = a$ and $y = 0$: $0 \cdot y_1 = 2a(a + x_1)$ $0 = 2a^2 + 2ax_1$ $2ax_1 = -2a^2$ $x_1 = -a$. Thus,the locus of the pole $(x_1, y_1)$ is $x = -a$,which is the equation of the directrix.

The locus of the poles of focal chords of a parabola is:

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