Question : If the radii of two circles are 6 cm and 3 cm and the length of the transverse common tangent is 8 cm, then the distance between the two centres is:
Option 1: $\sqrt{145} \text{ cm}$
Option 2: $\sqrt{140} \text{ cm}$
Option 3: $\sqrt{150} \text{ cm}$
Option 4: $\sqrt{135} \text{ cm}$
Correct Answer: $\sqrt{145} \text{ cm}$
Solution :
The distance between the centres of two circles with radii $r_1$ and $r_2$ (with $r_1 > r_2$), and the length of the transverse common tangent $t$,
$d = \sqrt{t^2 + (r_1 + r_2)^2}$
Substituting the given values into the equation:
$d = \sqrt{8^2 + (6 + 3)^2} = \sqrt{64 + 81} = \sqrt{145} \text{ cm}$
Hence, the correct answer is $\sqrt{145} \text{ cm}$.
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