The required coordinates are (6, 5).
Let A denotes the point (4, 3), and B denotes the point (9, 8).
Let C denote the point which divides the line segment AB in the ratio of 2:3.
The distance between A(4, 3) and B(9, 8) is given by
AB = √{(x₂ - x₁)² + (y₂ -y₁)²}
= √{(9 - 4)² + (8 - 3)²
= √{5² + 5²}
= √(2×5²)
= 5√2
We have AC + BC = AB
AC and BC are in the ratio 2:3.
Therefore 2x + 3x = AB
⇒5x = 5√2
⇒x = √2.
Therefore, AC = 2√2 and BC = 3√2.
Using the distance formula,
AC = √{(x - 4)² + (y - 3)²}
⇒ 2√2 = √{(x - 4)² + (y - 3)²}
⇒ √(2² + 2²) = √{(x - 4)² + (y - 3)²} (taking the 2 inside the square root.)
⇒ x - 4 = 2 ⇒ x = 6
⇒ y - 3 = 2 ⇒ y = 5.
Therefore the required coordinates are (6, 5).
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