(i) Write down the coordinates of 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) By section formula coordinates of P are,

$\Rightarrow \Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big) \\[1em] \Rightarrow \Big(\dfrac{1 \times 17 + 2 \times -4}{1 + 2}, \dfrac{1 \times 10 + 2 \times 1}{1 + 2}\Big) \\[1em] \Rightarrow \Big(\dfrac{17 - 8}{3}, \dfrac{10 + 2}{3}\Big) \\[1em] \Rightarrow \Big(\dfrac{9}{3}, \dfrac{12}{3}\Big) \\[1em] \Rightarrow (3, 4).

Hence, the coordinates of P are (3, 4).

(ii) By distance formula

$\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

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

Hence, the length of OP is 5 units.

(iii) Let AB be divided by the y-axis in the ratio m : n.

By section formula,

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

Thus, the ratio in which the y-axis divide the line AB is 4 : 17.