If the abscissa of a point P is 2, find the ratio in which this point divides the line segment joining the points (-4, 3) and (6, 3). Also, find the co-ordinates of point P.

Let point P be (2, y) and ratio in which it divides line segment joining the points (-4, 3) and (6, 3) be m₁ : m₂.

By section formula,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2}

Substituting values we get,

$\Rightarrow 2 = \dfrac{m$1 \times 6 + m2 \times -4}{m1 + m2} \\[1em] \Rightarrow 2m1 + 2m2 = 6m1 - 4m2 \\[1em] \Rightarrow 2m2 + 4m2 = 6m1 - 2m1 \\[1em] \Rightarrow 6m2 = 4m1 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{6}{4} = \dfrac{3}{2}.

m₁ : m₂ = 3 : 2.

$y = \dfrac{m$1y2 + m2y1}{m1 + m2}

Substituting values we get,

$\Rightarrow y = \dfrac{3 \times 3 + 2 \times 3}{3 + 2} \\[1em] \Rightarrow y = \dfrac{9 + 6}{5} \\[1em] \Rightarrow y = \dfrac{15}{5} = 3.$

P = (2, y) = (2, 3).

Hence, ratio = 3 : 2 and co-ordinates of P = (2, 3).