Question

The common chord of two intersection circles $c_1\&c_2$ can be seen from their centres at the angles of ${90}^{\circ }\&{60}^{\circ },$ respectively. If the distance between their centres is equal to $\sqrt{3}+1$, then the radii of $c_1\&c_2$ are

• A. $\sqrt{3}\&3$
• B. $\sqrt{2}\&2\sqrt{2}$
• C. $\sqrt{2}\&2$
• D. $2\sqrt{2}\&4$

Answer 1

The correct option is C $\sqrt{2}\&2$

Let $d$ be the distance between $O_1{O}_2$

In $△AB{O}_1$

$\angle A{O}_{1}B=\frac{1}{2}\angle A{O}_{1}C={45}^{\circ }$

$\tan 45=\frac{AB}{O_{1}B}⟹AB=x$

$r_1^2=x^2+x^2$

$r_1=x\sqrt{2}$

In $△A{O}_{2}B$

$\angle A{O}_{2}B=\frac{1}{2}\angle AOC={30}^{\circ }$

$\tan 30=\frac{AB}{O{B}_2}⟹AB=\frac{(d–x)}{\sqrt{3}}$

$⟹\frac{(d–x)}{\sqrt{3}}=x⟹x(\sqrt{3}+1)+\sqrt{3}+1$

$x=1$

$r_1=x\sqrt{2}=\sqrt{2}$

$r_2^2=x^2+(d-x{)}^2=1+3=4$

$r_2=2$