Question : The radius of two circles is 3 cm and 4 cm. The distance between the centres of the circles is 10 cm. What is the ratio of the length of the direct common tangent to the length of the transverse common tangent?
Option 1: $\sqrt{51}:\sqrt{68}$
Option 2: $\sqrt{33}:\sqrt{17}$
Option 3: $\sqrt{66}:\sqrt{51}$
Option 4: $\sqrt{28}:\sqrt{17}$
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Correct Answer: $\sqrt{33}:\sqrt{17}$
Solution :
The length of the transverse common tangent,
$=\sqrt{d^2 - (r_1 + r_2)^2}$, where $d$ is distance between centres and $r_1, r_2$ are radii of two circles.
$=\sqrt{10^2 - (3 + 4)^2}=\sqrt{100 - 49}=\sqrt{51}\;\mathrm{cm}$
The length of the direct common tangent,
$=\sqrt{d^2 - (r_1 - r_2)^2}$
$=\sqrt{10^2 - (3 - 4)^2}=\sqrt{100 - 1}=\sqrt{99}\;\mathrm{cm}$
The ratio of the length of the direct common tangent to the length of the transverse common tangent,
$=\sqrt{99}:\sqrt{51}=\sqrt{33}:\sqrt{17}$
Hence, the correct answer is $\sqrt{33}:\sqrt{17}$.
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