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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