In \DeltaPAB, A and B are fixed points lying on the x-axis and P is a variable point. y = m (x + 2) and 2mx + y = 4m are the internal angle bisectors of angle A and angle B respectively. Then which of the following must be correct?

Engineering

Mathematics

Standard Hyperbola

Question

In $∆$ PAB, A and B are fixed points lying on the x-axis and P is a variable point. y = m (x + 2) and 2mx + y = 4m are the internal angle bisectors of angle A and angle B respectively. Then which of the following must be correct?

Eccentricity of locus of point P is 1/3.

locus of point P is an ellipse.

Locus of point P is a hyperbola.

Eccentricity of locus of point P is 3.

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$\tan \frac{A}{2} = m; \tan \frac{B}{2} = 2 n$

$\frac{\tan \frac{B}{2}}{\tan \frac{A}{2}} = 2 \Rightarrow \frac{\sqrt{\frac{(s - a) (s - c)}{s (s - b)}}}{\sqrt{\frac{(s - b) (s - c)}{s (s - a)}}} = 2$

$\Rightarrow \frac{s - a}{s - b} = 2 \Rightarrow \frac{b - a}{c} = \frac{1}{3}$

(a = PB ; b = PA and C = AB = 4)

$\Rightarrow b - a = \frac{4}{3}$

$e = \frac{4}{4 / 3} = 3 Ans .$