A line segment joining A $\Big(-1, \dfrac{5}{3}\Big)$ and B(a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects the y-axis.

(i) Calculate the value of 'a'.

(ii) Calculate the co-ordinates of 'P'.

(i) Since, P is the point where the line segment AB intersects the y-axis.

Let P = (0, y).

Since, P divides AB in the ratio 1 : 3.

$\therefore 0 = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] \Rightarrow 0 = \dfrac{1 \times a + 3 \times -1}{1 + 3} \\[1em] \Rightarrow 0 = \dfrac{a - 3}{4} \\[1em] \Rightarrow 0 = a - 3 \\[1em] \Rightarrow a = 3. \\[1em]

Hence, a = 3.

(ii) By section formula,

$y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] = \dfrac{1 \times 5 + 3 \times \dfrac{5}{3}}{1 + 3} \\[1em] = \dfrac{5 + 5}{4} \\[1em] = \dfrac{10}{4} = \dfrac{5}{2} = 2\dfrac{1}{2}.

P = (0, y) = $\Big(0, 2\dfrac{1}{2}\Big)$.

Hence, co-ordinates of P = $\Big(0, 2\dfrac{1}{2}\Big)$.