Question : A direct common tangent is drawn to two circles of radius 25 cm and 20 cm. The centres of the two circles are 35 cm apart. What is the length (in cm) of the tangent?
Option 1: $25 \sqrt{2}$
Option 2: $25 \sqrt{3}$
Option 3: $20 \sqrt{3}$
Option 4: $20 \sqrt{2}$
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Correct Answer: $20 \sqrt{3}$
Solution :
The length of the direct common tangent ($D$) between two circles, 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.
$D=\sqrt{d^2 - (r_1 - r_2)^2}$
Given that \(d \) = 35 cm, \(r_1\) = 25 cm, and \(r_2\) = 20 cm
$D=\sqrt{(35)^2 - (25 - 20)^2} = \sqrt{1225 - 25} = \sqrt{1200} = 20\sqrt{3}$ cm
Hence, the correct answer is $20\sqrt{3}$.
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