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 = 200 and the sum of the squares of their radii is 100, what is the value of d?

Correct option is B

​Given :

Distance between centers of two circles = d

Length of direct common tangent = L

Length of transverse common tangent = M 

Relation:
​$$L^2 + M^2 = 200$$​​

Sum of squares of radii:
​$$r_1^2 + r_2^2 = 100$$​​

Formula Used :

Direct Common Tangent

​$$L^2 = d^2 - (r_1 - r_2)^2$$​​

Transverse Common Tangent

​$$M^2 = d^2 - (r_1 + r_2)^2$$​​

Key Identity

​$$(r_1 - r_2)^2 + (r_1 + r_2)^2 = 2(r_1^2 + r_2^2)$$​​
Solution :

Add the two tangent formulas:

L$$^2 + M^2 = \left[d^2 - (r_1 - r_2)^2\right] + \left[d^2 - (r_1 + r_2)^2\right]$$​​

​$$L^2 + M^2 = 2d^2 - \left[(r_1 - r_2)^2 + (r_1 + r_2)^2\right]$$​​

Use the identity:

​$$(r_1 - r_2)^2 + (r_1 + r_2)^2 = 2(r_1^2 + r_2^2)$$​​

Given $$r_1^2 + r_2^2 = 100:$$​​

​$$(r_1 - r_2)^2 + (r_1 + r_2)^2 = 2 \times 100 = 200$$​​

Thus

​$$L^2 + M^2 = 2d^2 - 200$$​​

Given ( L$$^2 + M^2 = 200 )$$​:

200 =$$2d^2 - 200$$​​

​$$2d^2 = 400$$​​

​$$d^2 = 200$$​​

d $$= \sqrt{200} = 10\sqrt{2}$$​​