Question

Point C and D divide segment AB respectively internally and externally, in the ratio m:n then B divides CD internally in the ratio --------

• A. \((m+n):m\)
• B. \((m+n):n\)
• C. \((m-n):(m+n)\)
• D. \((m+n):(m-n)\)

Hint:

Take convenient coordinates for A and B and find C and D using internal and external section formulae.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The points C and D divide AB internally and externally in the same ratio, so they are harmonic conjugates.

Step 2: Choose coordinates
Let \(A=0\) and \(B=1\) on a line. Then \(C=\dfrac{m}{m+n}\) and \(D=\dfrac{m}{m-n}\).

Step 3: Find the ratio CB : BD
\(CB=1-\dfrac{m}{m+n}=\dfrac{n}{m+n}\) and \(BD=\dfrac{m}{m-n}-1=\dfrac{n}{m-n}\).

Step 4: Divide
\(\dfrac{CB}{BD}=\dfrac{n}{m+n}\times\dfrac{m-n}{n}=\dfrac{m-n}{m+n}\)

Final Answer:
B divides CD internally in the ratio (m - n) : (m + n). \[ \boxed{\text{(C) }(m-n):(m+n)} \]