The radius of two circles are 3 cm and 4 cm. The distance between the circles is 10 cm. What is the ratio of the length of direct common tangent to the length of the transverse common tangent?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sqrt{33}:\sqrt{17}\)
For two circles with radii \(r_1\), \(r_2\) and distance \(d\) between their centres, each common tangent is a side of a right-angled triangle whose hypotenuse is \(d\):
Direct common tangent \(=\sqrt{d^2-(r_1-r_2)^2}\)
Transverse common tangent \(=\sqrt{d^2-(r_1+r_2)^2}\)
With \(d=10\), \(r_1=4\), \(r_2=3\):
Direct \(=\sqrt{100-1}=\sqrt{99}\)
Transverse \(=\sqrt{100-49}=\sqrt{51}\)
Ratio \(=\sqrt{99}:\sqrt{51}=\sqrt{33}:\sqrt{17}\) (both divided by \(\sqrt{3}\))