(i) Write down the co-ordinates of the point P that divides the line joining A(-4, 1) and B(17, 10) in the ratio 1 : 2.

(ii) Calculate the distance OP, where O is the origin.

(iii) In what ratio does the y-axis divide the line AB ?

(i) Let the co-ordinates of P be (x, y)

$\therefore x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] = \dfrac{1 \times 17 + 2 \times -4}{1 + 2} \\[1em] = \dfrac{17 + (-8)}{3} \\[1em] = \dfrac{9}{3} = 3.

and,

$y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] = \dfrac{1 \times 10 + 2 \times 1}{1 + 2} \\[1em] = \dfrac{10 + 2}{3} \\[1em] = \dfrac{12}{3} = 4.

P = (x, y) = (3, 4).

Hence, co-ordinates of point P = (3, 4).

(ii) Distance between two points = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

$OP = \sqrt{(3 - 0)^2 + (4 - 0)^2} \\[1em] = \sqrt{(3)^2 + (4)^2} \\[1em] = \sqrt{9 + 16} \\[1em] = \sqrt{25} \\[1em] = 5 \text{ units}.$

Hence, OP = 5 units.

(iii) Let point Q (0, z) on y-axis divide line AB in ratio m₁ : m₂.

By section formula,

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

Substituting values we get,

$\Rightarrow 0 = \dfrac{m$1 \times 17 + m2 \times -4}{m1 + m2} \\[1em] \Rightarrow 0 = 17m1 - 4m2 \\[1em] \Rightarrow 4m2 = 17m1 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{4}{17}.

m₁ : m₂ = 4 : 17.

Hence, ratio in which the y-axis divide the line AB = 4 : 17.