JEE Advance - Mathematics (2009 - Paper 1 Offline - No. 11)
If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then
$$b+c=4a$$
$$b+c=2a$$
locus of point $$A$$ is an ellipse
locus of point $$A$$ is a pair of straight lines
Explanation
From this given data, we can write as
$$2\cos \left( {{{B + C} \over 2}} \right)\cos \left( {{{B - C} \over 2}} \right) = 4{\sin ^2}{A \over 2}$$
$$\cos \left( {{{B - C} \over 2}} \right) = 2\sin (A/2)$$
$$ \Rightarrow {{\cos \left( {{{B - C} \over 2}} \right)} \over {\sin A/2}} = 2$$
$$ \Rightarrow {{\sin B + \sin C} \over {\sin A}} = 2$$
$$ \Rightarrow b + c = 2a$$
where a is a constant. Therefore, the locus of point A is an ellipse.
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