Question

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?

• A. 10
• B. 10√2
• C. 5√2
• D. 20

Answer 1

Answer:

B. 10√2

Let radii be (R) and (r), and distance between centers be (d).

For common tangents:

• Direct tangent:
  [
  L^2 = d^2 - (R - r)^2
  ]

• Transverse tangent:
  [
  M^2 = d^2 - (R + r)^2
  ]

---

Step 1: Add both equations

[
L^2 + M^2 = 2d^2 - [(R - r)^2 + (R + r)^2]
]

Use identity:
[
(R - r)^2 + (R + r)^2 = 2(R^2 + r^2)
]

So:
[
L^2 + M^2 = 2d^2 - 2(R^2 + r^2)
]

---

Step 2: Substitute given values

[
200 = 2d^2 - 2(100)
]

[
200 = 2d^2 - 200
]

---

Step 3: Solve

[
400 = 2d^2
]

[
d^2 = 200
]

[
d = \sqrt{200} = 10\sqrt{2}
]

---

✅ Final Answer:

[
\boxed{10\sqrt{2}}
]