Question

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 1

Hint: The length of a direct (external) common tangent to two circles is $ \sqrt{{{d}^{2}}-{{\left( {{r}_{1}}-{{r}_{2}} \right)}^{2}}} $ (Pythagoras' theorem), where $ d $ is the distance between the centers of the circles, and $ {{r}_{1}} $ and $ {{r}_{2}} $ are the radii of the given circles.
The length of a transverse (internal) common tangent to two circles is $ \sqrt{{{d}^{2}}-{{\left( {{r}_{1}}+{{r}_{2}} \right)}^{2}}} $ (Pythagoras' theorem), where $ d $ is the distance between the centers of the circles, and $ {{r}_{1}} $ and $ {{r}_{2}} $ are the radii of the given circles.
Form two equations. We cannot find the values of $ {{r}_{1}} $ and $ {{r}_{2}} $ , but their product $ {{r}_{1}}\times {{r}_{2}} $ can be determined.

Complete step by step answer:
The two tangents be $ AB=11 $ and $ PQ=7 $ , the radii be $ {{r}_{1}}=x $ and $ {{r}_{2}}=y $ , and $ d $ be the distance between the two circles.

Using the formula for the length of the direct common tangent, we have the following equation:
 $ A{{B}^{2}}={{d}^{2}}-{{\left( x-y \right)}^{2}} $
⇒ $ {{11}^{2}}={{d}^{2}}-\left( {{x}^{2}}+{{y}^{2}}-2xy \right) $
⇒ $ 121={{d}^{2}}-{{x}^{2}}-{{y}^{2}}+2xy $ ... (1)
Using the formula for the length of the transverse common tangent, we have the following equation:
 $ P{{Q}^{2}}={{d}^{2}}-{{\left( x+y \right)}^{2}} $
⇒ $ {{7}^{2}}={{d}^{2}}-\left( {{x}^{2}}+{{y}^{2}}+2xy \right) $
⇒ $ 49={{d}^{2}}-{{x}^{2}}-{{y}^{2}}-2xy $ ... (2)
Subtracting equation (2) from equation (1), we get:
 $ 121-49=\left( {{d}^{2}}-{{x}^{2}}-{{y}^{2}}+2xy \right)-\left( {{d}^{2}}-{{x}^{2}}-{{y}^{2}}-2xy \right) $
⇒ $ 72=4xy $
Dividing both sides by 4, gives us:
⇒ $ xy=18 $

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 $ y=mx+c $ is tangent to a circle $ {{x}^{2}}+{{y}^{2}}={{r}^{2}} $ if $ {{c}^{2}}={{r}^{2}}(1+{{m}^{2}}) $ .