Question

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

• A. a straight line
• B. a parabola
• C. an ellipse
• D. a hyperbola

Answer 1

The correct option is D a hyperbola

Let the coordinates of point $M$ be $(h,k)$ and $\angle MAB=\theta$

From $△MDA$
$\tan \theta =\frac{k}{h+1}...\left(1\right)$
Now, from $△MDB$
$\tan 2\theta =\frac{k}{2-h}...\left(2\right)$

As we know,
$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$

From $\left(1\right)$ and $\left(2\right)$,
$\frac{k}{2-h}=\frac{2.\frac{k}{h+1}}{1-{\left(\frac{k}{h+1}\right)}^2}$

$\Rightarrow \frac{k}{2-h}=\frac{2k(h+1)}{h^2+1+2h-k^2}$

$\Rightarrow h^2+1+2h-k^2=2h-2h^2+4$

$\Rightarrow 3h^2-k^2=3$

$\Rightarrow \frac{h^2}{1}-\frac{k^2}{3}=1$ which represents a hyperbola.