A circle with radius x touches another circle with radius 2x externally. What is the length of a direct common tangent?

Correct option is C

Given :

Radius of first circle = x
Radius of second circle = 2x
The circles touch externally.

Formula Used :
Length of direct common tangent between two circles:
T = $$\sqrt{d^2 - (r_1 - r_2)^2}$$​​
where d = distance between centers.

Solution :

Since circles touch externally:
d = $$r_1 + r_2 = x + 2$$​x = 3x

Substitute values:
T = $$\sqrt{(3x)^2 - (2x - x)^2}$$​​
=$$\sqrt{9x^2 - x^2}$$​​
=$$\sqrt{8x^2}$$​​
=$$2\sqrt{2}$$​x

​Length of the direct common tangent = $$2\sqrt{2}x$$​​

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