Question : 8 cm and 5 cm are the radii of two circles. If the distance between the centres of the two circles is 11 cm, then the length (in cm) of the common tangent of the two circles is:
Option 1: $2 \sqrt{7}$
Option 2: $3 \sqrt{7}$
Option 3: $\sqrt{7}$
Option 4: $4\sqrt{7}$
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Correct Answer: $4\sqrt{7}$
Solution :
Given: The distance between the centres of two circles having radii 8 cm and 5 cm, is 11cm.
Let the radius of the bigger circle ($r_1$) = 8 cm,
And the radius of the smaller circle ($r_2$) = 5 cm
Distance between the centres of two circles ($d$) = 11 cm
We know,
The length of the direct tangent ($l$) = $\sqrt{d^2-(r_1-r_2)^2}$
⇒ $l=\sqrt{11^2-(8-5)^2}$
⇒ $l=\sqrt{11^2-(3)^2}$
⇒ $l=\sqrt{121-9}$
$\therefore l=\sqrt{112}=4\sqrt{7}$ cm
Hence, the correct answer is $4\sqrt{7}$.
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