The distance between two circle centers is d. The lengths of direct and transverse common tangents are L and M respectively. If L² + M² = 320 and sum of squares of radii is 160, what is d?

Question

Accepted Answer

1. For two circles with radii $r_1,r_2$ and centre distance $d$:

- **Direct tangent:** $L^2=d^2-(r_1-r_2)^2$
   - **Transverse tangent:** $M^2=d^2-(r_1+r_2)^2$

2. Add them: $L^2+M^2=2d^2-[(r_1-r_2)^2+(r_1+r_2)^2]=2d^2-2(r_1^2+r_2^2)$.

3. Plug values $L^2+M^2=320$ and $r_1^2+r_2^2=160$:

$320=2(d^2-160)\Rightarrow d^2=320$.

4. So $d=\sqrt{320}=8\sqrt{5}$. Choice (b).

FINAL_ANSWER: b

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