The distance between the centers of two circles is d. The lengths of their direct and transverse common tangents are L and M, respectively. If L2 + M2 = 338 and the sum of the squares of their radii is r12 + r22 = 162, what is the value of d?
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Correct answer: B
Explanation
The correct answer is option 2 i.e. .
For two circles with radii r1 and r2 and center distance d:
Direct common tangent, L2 = d2 − (r1 − r2)2
And Transverse common tangent, M2 = d2 − (r1 + r2)2
Add the two equations, L2 + M2 = d2 − (r1 − r2)2 + d2 − (r1 + r2)2
⇒ 338 = 2d2 - (r12 + r22 - 2r1r2 + r12 + r22 + 2r1r2)
⇒ 338 = 2d2 - 2(r12 + r22 )
⇒ 338 = 2d2 - 2(162)
⇒ 2d2 = 338 + 324
⇒ d2 = 662/2
⇒ d =