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

sania bajaj , 12 Years ago

Grade Upto college level

1 Answers

Jitender Singh

Last Activity: 11 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

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