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Question

# Let A(−1,0) and B(2,0) be two points. A point M moves in the plane in such a way that ∠MBA=2∠MAB. Then the point M moves along

A
a straight line
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B
a parabola
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C
an ellipse
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D
a hyperbola
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Solution

## The correct option is D a hyperbolaLet the coordinates of point M be (h,k) and ∠MAB=θ From △MDA tanθ=kh+1 ...(1) Now, from △MDB tan2θ=k2−h ...(2) As we know, tan2θ=2tanθ1−tan2θ From (1) and (2), k2−h=2 .kh+11−(kh+1)2 ⇒k2−h=2k(h+1)h2+1+2h−k2 ⇒h2+1+2h−k2=2h−2h2+4 ⇒3h2−k2=3 ⇒h21−k23=1 which represents a hyperbola.

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