Question : The distance between the centres of two circles of radii 3 cm and 8 cm is 13 cm. If the points of contact of a direct common tangent to the circles are P and Q, then the length of the line segment PQ is:
Option 1: 11.9 cm
Option 2: 12 cm
Option 3: 11.58 cm
Option 4: 11.5 cm
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Correct Answer: 12 cm
Solution :
Given: Two circles of radii 3 cm and 8 cm, the distance between the centres = 13 cm and PQ is the direct common tangent.
We know,
The length of the direct common tangent of two circles = $\sqrt{d^2-(r_1-r_2)^2}$
Here, the distance between the centres($d$) = 13 cm, radius($r_1$) = 8 cm and radius($r_2$) = 3 cm
So, the length of the direct common tangent(PQ)
= $\sqrt{13^2-(8-3)^2}$
= $\sqrt{13^2-5^2}$
= $\sqrt{144}$
= $12\ \text{cm}$
Hence, the correct answer is 12 cm.
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