A `(-1, 0) and B (2, 0)` are two given points. A point M is moving in such a way thatthe angle B in the triangle AMB remains twice as large as the angle A. Show that the locus of the pointM is a hyperbola. Find the eccentricity of the hyperbola.

Options

A. eccentricity equals 2
B. vertices `(+-3,0)`
C. length of latus-rectum equals 6
D. equation of directrices are `x=+-(1)/(2)`

Explanation

Given, `theta= 2 phi rArr tan theta 2 phi`
`rArr tan theta = (2 tan phi)/(1-tan^(2) phi)rArr (-y_(0))/(x_(0)-2)=(2y_(0)(x_(0)+1))/((x_(0)+1)^(2)-y_(0)^(2))rArr (-1)/(x_(0)-2)=(2(x_(0)+1))/((x_(0)+1)^(2)-y_(0)^(2))`
`rArr 3x_(0)^(2)-y_(0)^(2)=3`
`:.` Locus of M is hyperbola `3x^(2)-y^(2)=3`
Now, verify alternatives.