Points A(4, 2) and B(8, 4) are given. Does the point P(2, 1) lie on the segment AB?

Class 10Coordinate GeometryGeometry — Coordinates

Step-by-Step Solution

Line AB has slope

\frac{4 - 2}{8 - 4} = \frac{1}{2}

and equation

y = \frac{x}{2}

. P(2, 1) satisfies it, so P is on the line AB. But a point of the segment must have

x

between 4 and 8, and P has

x = 2

, so P lies outside the segment (beyond A).

Answer: A) No — P lies on line AB but outside the segment

**Section formula (internal)**:
A point

P

dividing segment

A B

in ratio

m : n (P

closer to A means

m

smaller) has coordinates:

P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

Where

A = (x

₁,

y

₁),

B = (x

₂,

y

₂).

**Midpoint** is the special case

m = n : M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

.

**Example**: Find the point dividing

A B

in ratio 2:3 where

A = (1, 5), B = (6, - 5) . x = \frac{2 \cdot 6 + 3 \cdot 1}{5} = \frac{15}{5} = 3. y = \frac{2 \cdot ( - 5 ) + 3 \cdot 5}{5} = \frac{5}{5} = 1

.
Point: (3, 1).

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