Question : Two circles of radii 8 cm and 3 cm, respectively, are 13 cm apart. AB is a direct common tangent touch to both the circles at A and B respectively then the length of AB is:
Option 1: 10 cm
Option 2: 12 cm
Option 3: 8 cm
Option 4: 6 cm
Correct Answer: 12 cm
Solution :
The length of the direct common tangent AB, where $d$ is the distance between the centres of the two circles, and $r_1$ and $r_2$ are the radii of the two circles.
$AB = \sqrt{d^2 - (r_1 - r_2)^2}$
Substituting the given values,
$AB = \sqrt{(13)^2 - (8 - 3)^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}$
Hence, the correct answer is 12 cm.
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