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

Answer: 0.5

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),

$C_{1}A=C_{2}A=r=OD=2$
and $C_1\equiv (2,h)$
where $h^2=A{C}_1^2-A{D}^2=4-1=3$
or $C_1\equiv (2,\sqrt{3}),C_2\equiv (2,-\sqrt{3})$
If $\angle C_{1}A{C}_2=\theta$, then
$cos\theta =\frac{A{C}_1^2+A{C}_2^2-C_1{C}_2^2}{2A{C}_1\times A{C}_2}=\frac{1}{2}$