Question

Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circles is

• A. $\sqrt{r}$
• B. $\sqrt{2}rAB$
• C. $\sqrt{3}r$
• D. $\frac{\sqrt{3}}{2}r$

Answer 1

The correct option is C

$\sqrt{3}r$

Let r be the radius of each circle.

$\therefore OA=r,OE=\frac{1}{2}OC=\frac{1}{2}r$

$\therefore E{A}^2=O{A}^2-O{E}^2$

$=(r{)}^2-{\left(\frac{1}{2}r\right)}^2=r^2-\frac{1}{4}{r}^2$

$=\frac{3}{4}{r}^2$

$\therefore EA=\frac{\sqrt{3}}{2}r$

$andAB=2\times EA=2\times \frac{\sqrt{3}}{2}r=\sqrt{3}r$