(i) Write down the co-ordinates of the point P that divides the line segment 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) The point P divides the line segment joining A(-4, 1) and B(17, 10) in the ratio m₁ : m₂ = 1 : 2.

Let coordinates of P be (x, y).

By section-formula,

(x, y) = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

$\Rightarrow (x, y) = \Big(\dfrac{1(17) + 2(-4)}{1 + 2}, \dfrac{1(10) + 2(1)}{1 + 2}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{17 - 8}{3}, \dfrac{10 + 2}{3}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{9}{3}, \dfrac{12}{3}\Big) \\[1em] \Rightarrow (x, y) = (3, 4).$

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

Hence, coordinates of P = (3, 4).

(ii) Using distance formula,

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

Substitute values we get,

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

Hence, distance of OP is 5 units.

(iii) A point on the y-axis has an x-coordinate of 0. Let the y-axis cut AB at K(0, y). let the ratio be k : 1.

By section-formula,

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

Substitute values we get,

$\Rightarrow 0 = \Big(\dfrac{k(17) + 1(-4)}{k + 1}\Big) \\[1em] \Rightarrow 17k - 4 = 0 \\[1em] \Rightarrow 17k = 4 \\[1em] \Rightarrow k = \dfrac{4}{17} \\[1em] \Rightarrow k : 1 = \dfrac{4}{17} : 1 = 4 : 17.$

Hence, y-axis cut line AB in ratio 4 : 17.