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# I need image of locus of the poles of chords of parabola which subtend a constant angle at the vertex

Jitender Singh IIT Delhi
6 years ago
Ans:
Let the parabola be:
$y^{2} = 4ax$
Let t1& t2be parametric points of two focal chords on the parabola. Let (h, k) be the coordinates of the poles.
The coordinates of the point P & Q of the focal chords on the parabola are:
$P(at_{1}^{2}, 2at_{1}), Q(at_{2}^{2}, 2at_{2})$
$h = at_{1}t_{2}$
$k = a(t_{1}+t_{2})$
Letµ be the angle focal chords subtend at vetex O(0, 0).
Slope of OP:
$\frac{2}{t_{1}}$
Slope of OQ:
$\frac{2}{t_{2}}$
Then,
$tan\mu = \frac{(\frac{2}{t_{1}}-\frac{2}{t_{2}})}{1+\frac{4}{t_{1}t_{2}}}$
$tan\mu = \frac{2(t_{2}-t_{1})}{t_{1}t_{2}+4}$
$tan^{2}\mu(t_{1}t_{2}+4)^{2} = 4.(t_{2}-t_{1})^{2} = 4.((t_{1}+t_{2})^{2}-4t_{1}t_{2})$
$tan^{2}\mu(\frac{h}{a}+4)^{2} = 4.((\frac{k}{a})^{2}-4.\frac{h}{a})$
$h\rightarrow x, k\rightarrow y$
$tan^{2}\mu(\frac{x}{a}+4)^{2} = 4.((\frac{y}{a})^{2}-4.\frac{x}{a})$
Thanks & Regards
Jitender Singh
IIT Delhi