If the abscissa of a point P is 2, find the ratio in which it divides the line segment joining the points (-4, 3) and (6, 3). Hence, find the coordinates of P.

Let coordinates of A be (-4, 3) and of B be (6, 3) and of P be (2, y).

Let the ratio in which the P divides AB be m : n.

By section formula,

x-coordinate = $\dfrac{mx$2 + nx1}{m+ n}

$\Rightarrow 2 = \dfrac{m \times 6 + n \times (-4)}{m + n} \\[1em] \Rightarrow 2 = \dfrac{6m - 4n}{m + n} \\[1em] \Rightarrow 2(m + n) = 6m - 4n \\[1em] \Rightarrow 2m + 2n = 6m - 4n \\[1em] \Rightarrow 2n + 4n = 6m - 2m \\[1em] \Rightarrow 6n = 4m \\[1em] \Rightarrow 6n = 4m \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{6}{4} \\[1em] \Rightarrow m : n = 3 : 2.$

Similarly for y-coordinate,

$\Rightarrow y = \dfrac{my$2 + ny1}{m + n} \\[1em] = \dfrac{3 \times 3 + 2 \times 3}{3 + 2} \\[1em] = \dfrac{9 + 6}{5} \\[1em] = \dfrac{15}{5} \\[1em] = 3.

Hence, the coordinates of P is (2, 3) and it divides the line in the ratio 3 : 2.