
If the length of a common internal tangent to two circles is 7 and that of a common external tangent is 11, then the product of the radii of the two circles is:
A. 36
B. 9
C. 18
D. 4
Answer
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Hint: The length of a direct (external) common tangent to two circles is (Pythagoras' theorem), where is the distance between the centers of the circles, and and are the radii of the given circles.
The length of a transverse (internal) common tangent to two circles is (Pythagoras' theorem), where is the distance between the centers of the circles, and and are the radii of the given circles.
Form two equations. We cannot find the values of and , but their product can be determined.
Complete step by step answer:
The two tangents be and , the radii be and , and be the distance between the two circles.
Using the formula for the length of the direct common tangent, we have the following equation:
⇒
⇒ ... (1)
Using the formula for the length of the transverse common tangent, we have the following equation:
⇒
⇒ ... (2)
Subtracting equation (2) from equation (1), we get:
⇒
Dividing both sides by 4, gives us:
⇒
Therefore, the answer is C. 18.
Note: Both the direct tangents are equal in length. Both the transverse tangents are also equal.
Direct tangents are longer than the transverse tangents.
The tangents to a circle are perpendicular to the radius of the circle at the point of contact.
A line is tangent to a circle if .
The length of a transverse (internal) common tangent to two circles is
Form two equations. We cannot find the values of
Complete step by step answer:
The two tangents be

Using the formula for the length of the direct common tangent, we have the following equation:
⇒
⇒
Using the formula for the length of the transverse common tangent, we have the following equation:
⇒
⇒
Subtracting equation (2) from equation (1), we get:
⇒
Dividing both sides by 4, gives us:
⇒
Therefore, the answer is C. 18.
Note: Both the direct tangents are equal in length. Both the transverse tangents are also equal.
Direct tangents are longer than the transverse tangents.
The tangents to a circle are perpendicular to the radius of the circle at the point of contact.
A line
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