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: Understanding the Concept:
The Section Formula is used to find the coordinates of a point that divides a line segment joining two given points in a specific ratio.

Step 2: Key Formula or Approach:
For points $(x_1, y_1)$ and $(x_2, y_2)$ divided in ratio $m:n$ internally, the coordinates $(x, y)$ are: \[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n} \]

Step 3: Detailed Explanation:
Given: $(x_1, y_1) = (1, 2)$, $(x_2, y_2) = (5, 6)$, and $m:n = 1:3$.
Calculating $x$: \[ x = \frac{1(5) + 3(1)}{1+3} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \] Calculating $y$: \[ y = \frac{1(6) + 3(2)}{1+3} = \frac{6 + 6}{4} = \frac{12}{4} = 3 \] The point is $(2, 3)$.

Step 4: Final Answer:
The point dividing the segment is (2, 3).

Answer 2

Rather than deriving the dividing point from scratch, each candidate point can be tested directly: if a point \(P\) truly divides segment \(AB\) with \(A=(1,2)\) and \(B=(5,6)\) in the ratio \(1:3\), then \(P\) must satisfy \(P = A + \dfrac{1}{4}(B-A)\), since dividing in ratio \(1:3\) internally places \(P\) exactly \(\dfrac{1}{1+3}=\dfrac{1}{4}\) of the way from \(A\) to \(B\).

\[ B - A = (5-1,\ 6-2) = (4,4), \qquad \frac{1}{4}(4,4) = (1,1), \qquad A + (1,1) = (2,3) \]

1. 2, 3: This is exactly the point obtained by moving \(\frac{1}{4}\) of the way from \(A\) to \(B\), matching the required ratio precisely.
2. 3, 4: Checking this against the ratio condition, moving from \(A=(1,2)\) to \((3,4)\) covers a displacement of \((2,2)\), which is \(\frac{1}{2}\) of the full displacement \((4,4)\), corresponding to a 1:1 ratio, not 1:3.
3. 4, 5: Moving from \(A\) to \((4,5)\) covers \((3,3)\), which is \(\frac{3}{4}\) of the full displacement, corresponding to a 3:1 ratio, the reverse of what's asked.
4. 2, 2: This point doesn't even lie on the line joining \(A\) and \(B\) (the segment has slope 1, so any point on it must have equal x and y offsets from \(A\); \((2,2)\) gives an x-offset of 1 but a y-offset of 0), so it fails even before checking the ratio.

Only one candidate correctly sits exactly one quarter of the way along the segment from \(A\) toward \(B\).

Therefore, the correct answer is (2, 3).