First slide

Equation of line in Straight lines

Question

Two straight lines rotate about two fixed points (-a, O) and (a,0) in anti clockwise direction. If they start from their position of coincidence such that one rotates at a rate double of the other, then locus of curve is

Moderate

A

$x^{2} + y^{2} + a x - 3 a^{2} = 0$
B

$x^{2} - y^{2} - 2 a x - 3 a^{2} = 0$
C

$x^{2} + y^{2} - 2 a x - 3 a^{2} = 0$
D

$x^{2} - y^{2} + 2 a x - 3 a^{2} = 0$

Solution

$Let P B make angle θ and P A make angle 2 θ with x -axis. Let P is (h, k) .$
$s l o p e \tan θ = \frac{k}{h + a} - - - - - (1)$
$\tan 2 θ = \frac{k}{h - a} - - - - (2)$
From (2)
$\frac{2 \tan θ}{1 - \tan^{2} θ} = \frac{k}{h - a}$
Locus on solving is
$\frac{2 k (h + a)}{{(h + a)}^{2} - k^{2}} = \frac{k}{h - a}$
$2 k (h^{2} - a^{2}) = k {(h + a)}^{2} - k^{3}$
$\begin{aligned} h^{2} + k^{2} - 2 a h - 3 a^{2} = 0 \\ x^{2} + y^{2} - 2 a x - 3 a^{2} = 0 \end{aligned}$

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