Question

The point dividing (1, 2) and (5, 6) in ratio 1:3 internally

• A. 2, 3
• B. 3, 4
• C. 4, 5
• D. 2, 2

Hint:

To avoid mixing up the numbers, remember the "cross-multiplication" rule: the first part of the ratio ($m$) multiplies the second point's coordinates, and the second part of the ratio ($n$) multiplies the first point's coordinates.

Answer 1

The Correct Option is A

Step 1: For point \(A=(1,2)\) and \(B=(5,6)\) with a point \(P\) dividing \(AB\) in ratio \(1:3\) internally, \(P\) lies at a fraction \(\dfrac{1}{1+3} = \dfrac{1}{4}\) of the way from \(A\) to \(B\).

Step 2: Find the displacement vector from \(A\) to \(B\): \(B - A = (5-1,\ 6-2) = (4,4)\).

Step 3: Scale this displacement by \(\dfrac{1}{4}\) and add it to \(A\): \(P = (1,2) + \dfrac{1}{4}(4,4) = (1,2)+(1,1)\).
\[ P = \boxed{(2,\ 3)} \]