Two common tangents AC and BD touch two equal circles each of radius 7 cm, at points A, C, B, and D, respectively, as shown in the figure. If the length of BD is 48 cm, what is the length of AC?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 50 cm.
The radius of each circle = 7 cm
BD is a transverse tangent on both circles
AC is a direct common tangent to both circles
Let x be the distance between the centers of the two circles
The length of direct common tangents = { (Distance between the centers of the two circles)2 - (Difference between the radius of two circles)2 }1/2
The length of direct transverse tangents = { (Distance between the centers of the two circles)2 - (Sum of the radius of two circles)2 }1/2
Here, BD = 48, The sum of the radius of the two circles = 14
Thus, BD = { (x)2 - (14)2 }1/2
⇒ BD2 = x2 - 196 ⇒ 482 + 196 = x2
⇒ x2 = 2304 + 196 = 2500
⇒ x = 50
The length of direct common tangents = { (x)2 - (7 - 7)2 }1/2
⇒ AC = ( 502 - 0)1/2
⇒ AC = (502)1/2 = 50 cm