Question

The circles which can be drawn to pass through $(1,0)$ and $(3 ,0)$ and to touch the $y$-axis intersect at an angle $\theta$. Then $cos\, \theta$ is equal to ______.

Answer 1

Let $A \equiv (1, 0), B \equiv (3, 0)$, and $C_1$ and $C_2$ be the centers of circles passing through $A$ and $B$ and touching the $y$ -axis at $P_1$ and $P_2$, respectively. If $r$ is the radius (here radii of both circles will be the same),

$C_1A = C_2A = r = OD = 2$
and $C_1 \equiv (2, h)$
where $h^2 = AC^2_1 - AD^2 = 4 - 1 = 3$
or $C_1 \equiv (2, \sqrt{3}), C_2 \equiv (2, -\sqrt{3})$
If $\angle C_1AC_2 = \theta$, then
$cos\,\theta =\frac{AC^2_1 + AC^2_2 - C_1C^2_2}{2AC_1 \times AC_2} = \frac{1}{2}$