Section formula: x = (m·x₂ + n·x₁)/(m+n). Internal division with full working.

Point P Divides AB in Ratio 2:3 — Section Formula Application

Question

Point P divides the line segment joining A(1, 2) and B(4, 8) internally in the ratio 2:3. Find the coordinates of P.

Solution — Step by Step

We have two points: A(1, 2) and B(4, 8). Point P divides AB internally in the ratio m:n = 2:3.

Label carefully: A is $(x_1, y_1)$ and B is $(x_2, y_2)$ . The ratio 2:3 means P is closer to A, since the smaller part comes first.

The internal division formula gives us:

x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \quad y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}

Notice the structure: the coordinate of the far point gets multiplied by m (the first ratio number). This trips up most students — we’ll come back to it.

Plugging in m = 2, n = 3, $x_1$ = 1, $x_2$ = 4:

x = \frac{2 \times 4 + 3 \times 1}{2 + 3} = \frac{8 + 3}{5} = \frac{11}{5}

Same formula, now for y. With $y_1$ = 2, $y_2$ = 8:

y = \frac{2 \times 8 + 3 \times 2}{2 + 3} = \frac{16 + 6}{5} = \frac{22}{5}

Point P has coordinates $\left(\dfrac{11}{5},\ \dfrac{22}{5}\right)$ .

Quick sanity check: $\frac{11}{5} = 2.2$ lies between $x_1 = 1$ and $x_2 = 4$ . ✓ And 2.2 is closer to 1 than to 4, matching the 2:3 ratio. ✓

Why This Works

The section formula is really a weighted average. When P divides AB in 2:3, it means AP:PB = 2:3 — so P has “moved” 2 parts toward B out of a total 5 parts. That’s exactly $\frac{2}{5}$ of the way from A to B.

The x-coordinate of P is: $x_A + \frac{2}{5}(x_B - x_A) = 1 + \frac{2}{5}(3) = 1 + \frac{6}{5} = \frac{11}{5}$ . Same answer, different path.

This weighted-average framing is why $x_2$ gets multiplied by m (not n) — B’s coordinate carries weight proportional to how far P has traveled toward it.

Alternative Method (Using Parametric Form)

We can reach P by starting at A and moving a fraction of the way to B.

Since the ratio is 2:3, P is $\frac{2}{5}$ of the way from A to B:

P = A + \frac{2}{5}(B - A)

x_P = 1 + \frac{2}{5}(4 - 1) = 1 + \frac{6}{5} = \frac{11}{5}

y_P = 2 + \frac{2}{5}(8 - 2) = 2 + \frac{12}{5} = \frac{22}{5}

This method is slower to write in an exam but excellent for building intuition — and useful when the ratio is given as a fraction rather than m:n.

Common Mistake

Swapping which point gets multiplied by m.

The formula is $\frac{m \cdot x_2 + n \cdot x_1}{m+n}$ — notice $x_2$ (the second point, B) is multiplied by m (the first ratio number). Students flip this and write $m \cdot x_1 + n \cdot x_2$ by thinking “m goes with the first point.” Wrong.

Memory trick: m pairs with the far side. If P divides AB in ratio AP:PB = m:n, then B is on the far side from A’s end — so B’s coordinate gets m.

In CBSE boards, always verify your answer. After finding P, check that the distance AP:PB equals 2:3 using the distance formula. One extra line in your solution and you lock in full marks even if the examiner is strict.

The section formula is a high-weightage topic in Class 10 — expect one direct application question and possibly a proof-based or centroid question in the same chapter. PYQs from 2022 and 2023 CBSE both had section formula as a 3-mark question in the same format as this one.

Question

Solution — Step by Step

Why This Works

Alternative Method (Using Parametric Form)

Common Mistake

Try These Next