question_answer

Two circles touch each other internally. Their radii are 4 cm and 6 cm. What is the length of the longest chord of the outer circle which is outside the inner circle?

A)  \[4\sqrt{2}\,cm\]                       

B)  \[4\sqrt{3}\,cm\]

C)  \[6\sqrt{3}\,cm\]                       

D)  \[8\sqrt{2}\,cm\]

Correct Answer: D

Solution :

Let \[{{C}_{1}}\] and \[{{C}_{2}}\] are the centres of the two circles having radii, 4 cm and 6 cm respectively and AB is the largest chord of circle \[{{C}_{2}}\] which touches inner circles \[{{C}_{1}}\] at 0. Distance \[{{C}_{1}}\,{{C}_{2}}=6-4=2\,cm\] Distance \[O{{C}_{2}}=2\,cm\] Now, In right angled triangle \[AC{{O}_{2}}\] \[AO=\sqrt{{{(A{{C}_{2}})}^{2}}-{{(O{{C}_{2}})}^{2}}}\] \[AO=\sqrt{{{6}^{2}}-{{2}^{2}}}\Rightarrow AO=4\sqrt{2}\] Chord\[AB=2AO=2\times 4\sqrt{2}=\mathbf{8}\sqrt{\mathbf{2}}\,\mathbf{cm}\]