In ` A B C ,` if `A B=c` is fixed, and `cosA+cosB+2cos C=2` then the locus of vertex `C` is ellipse (b) hyperbola (c) circle (d) parabola

Options

A. ellipse
B. hyperbola
C. circle
D. parabola

Answer

A

Explanation

`cos A + cos B = 2(1 - cos C) = 4 sin^(2).(C)/(2)`
`rArr 2cos.(A +B)/(2) cos.(A -B)/(2) = 4 sin^(2).(C)/(2)`
`rArr cos.(A -B)/(2) = 2 sin.(C)/(2)`
`rArr 2 cos.(C)/(2) cos.(A -B)/(2) = 4 sin.(C)/(2) cos.(C)/(2)`
`rArr 2 sin.(A + B)/(2) cos.(A -B)/(2) = 2 sin C`
`rArr sin A+ sin B = 2 sin C`
`rArr a + b = 2c`(constant)
So, locus of vertex C is an ellipse with vertices A and B as foci