The 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 y-axis. Calculate

(i) the value of a.

(ii) the coordinates of P.

(i) Let P(x, y) divide the line segment joining the points A$\Big(-1, \dfrac{5}{3}\Big)$, B(a, 5) in the ratio 1 : 3

We know that,

Section-formula = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big).

Putting values in above equation we get, coordinates of P

$= \Big(\dfrac{1 \times a + 3 \times (-1)}{1 + 3}, \dfrac{1 \times 5 + 3 \times \dfrac{5}{3}}{1 + 3}\Big) \\[1em] = \Big(\dfrac{a - 3}{4}, \dfrac{5 + 5}{4}\Big) \\[1em] = \Big(\dfrac{a - 3}{4}, \dfrac{10}{4}\Big) \\[1em] = \Big(\dfrac{a - 3}{4}, \dfrac{5}{2}\Big).$

According to the question,

The point P lies on y-axis so, x coordinate will be equal to zero.

$\therefore \dfrac{a - 3}{4} = 0$
⇒ a - 3 = 0
⇒ a = 3.

Hence, the value of a = 3.

(ii) From part (i) we get the y coordinate of P = $\dfrac{5}{2}$.

Hence, the coordinates of P are $\Big(0, \dfrac{5}{2}\Big).$