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

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. 36B. 9C. 18D. 4

Vedantu Content Team · Accepted Answer

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}}) $ .