Question : Two circles touch each other internally. Their radii are 3 cm and 4 cm. What is the length of the biggest chord of the circle with radii of 4 cm which is outside the inner circle?
Option 1: $5 \sqrt{3} \mathrm{~cm}$
Option 2: $6 \sqrt{3} \mathrm{~cm}$
Option 3: $4 \sqrt{3} \mathrm{~cm}$
Option 4: $2 \sqrt{3} \mathrm{~cm}$
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Correct Answer: $2 \sqrt{3} \mathrm{~cm}$
Solution : From the figure, MN is the diameter of the smaller circle and PQ is the biggest chord of the greater circle. O is the centre of the greater circle and O' is the centre of the inner circle. Now, OM = MN – ON = 6 – 4 = 2 cm Now, In $\triangle$PMO, PM = $\sqrt{\text{OP}^2-\text{OM}^2}$ ⇒ PM = $\sqrt{4^2-2^2}$ ⇒ PM = $\sqrt{16-4}$ ⇒ PM = $\sqrt{12}$ $\therefore$ PM = $2\sqrt{3}$ cm Hence, the correct answer is $2\sqrt{3}$ cm.
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