question_answer

Two concentric circles of radii a and b, where \[a>b,\]are given. The length of a chord of the larger circle which touches the other circle is                      

A) \[\sqrt{{{a}^{2}}-{{b}^{2}}}\]               

B)        \[2\sqrt{{{a}^{2}}-{{b}^{2}}}\]             

C)        \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]             

D)        \[2\sqrt{{{a}^{2}}+{{b}^{2}}}\]

Correct Answer: B

Solution :

In figure, AB is a chord of circle \[{{C}_{1}}\]which is a tangent to \[{{C}_{2}}\].                          Since, tangent is perpendicular to radius through point of contact             \[\therefore \] \[\angle OCA={{90}^{o}}\Rightarrow OA=a,\,\,OC=b\] In \[\Delta \,OCA,\] \[{{(OA)}^{2}}={{(OC)}^{2}}+{{(AC)}^{2}}\] \[\Rightarrow \]   \[{{a}^{2}}={{b}^{2}}+{{(AC)}^{2}}\Rightarrow AC=\sqrt{{{a}^{2}}-{{b}^{2}}}\] \[\therefore \] Length of chord \[AB=2AC=2\sqrt{{{a}^{2}}-{{b}^{2}}}.\]