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Locus of the poles of focal chords of a parabola is of parabola
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The directrix
Let equation of the parabola is $\mathrm{y}^2=4 \mathrm{ax}$ Polar of the pole $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ w.r.t the given parabola is $\mathrm{yy}_1-2 \mathrm{ax}-2 \mathrm{ax}_1=0$, which is identical to the focal chord and passes through focus $(a, 0)$ of the parabola, then
$0 \times y_1-2 a \times a-2 a x_1=0$
$-2 a^2-2 a x_1=0$
$\mathrm{x}_1=-\mathrm{a}$
$x=-a$, which is equation of the directrix. Therefore, the desired locus is directrix of the parabola.
$0 \times y_1-2 a \times a-2 a x_1=0$
$-2 a^2-2 a x_1=0$
$\mathrm{x}_1=-\mathrm{a}$
$x=-a$, which is equation of the directrix. Therefore, the desired locus is directrix of the parabola.
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