Two circles with radii r₁ and r₂ touch each other externally. If the length of their direct common tangent is T, which of the following is the correct relationship between T, r₁ and r₂ ?​

Correct option is C

Given :

Two circles with radii $$r_1$$​ and $$r_2$$​ touch each other externally.
Length of their direct common tangent = T.

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 the circles touch externally, the distance between their centers:
d = $$r_1 + r_2$$​​

Substitute in the formula:
T $$= \sqrt{(r_1 + r_2)^2 - (r_1 - r_2)^2}$$​​

Using identity:
​$$(a+b)^2 - (a-b)^2 = 4ab$$​​

T =$$\sqrt{4r_1 r_2}$$​​

T$$= 2\sqrt{r_1 r_2}$$​​