question_answer

If two circles are such that the centre of one lies on the circumference of the other then the ratio of the common chord of the two circles to the radius of any one of the circles is :

A) \[2:1\]                           

B) \[\sqrt{3}:1\]

C) \[\sqrt{5}:1\]                 

D) \[4:1\]

Correct Answer: B

Solution :

Here let O, O' be the centres of the circle. As the centre of each lies on the circumference of the other, the two circles will have the same ra- dius ame radius. Let it be r. \[\therefore \,\,\,OC=O'C=\frac{r}{2}\] \[\therefore \,\,\,AC=\sqrt{O{{A}^{2}}-O{{C}^{2}}}\] \[=\sqrt{{{r}^{2}}-\frac{{{r}^{2}}}{4}}=\frac{\sqrt{3}}{2}r\] \[AB=\sqrt{3\,\,}\,\,r\] Hence \[\frac{common\text{ }chord~}{radius}=\sqrt{3}\,\,r:r=\sqrt{3}:1\]