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

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

Grade:Upto college level

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
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.
241-2125_Parabola.gif
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
askIITians Faculty

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