7 min readPathway sheet2-mark and 3-mark questions

Finding the ratio of division

When the ratio is unknown, call it k : 1, write the section formula with k, and solve using the coordinate you know.

Unknown ratioTake it as k : 1, so P = ((kx₂ + x₁)/(k + 1), (ky₂ + y₁)/(k + 1))

Divided by the x-axisThe point of division has y = 0

Divided by the y-axisThe point of division has x = 0

For a point between A and Bk comes out positive

The k : 1 method

1
Let the required ratio be k : 1.
2
Write the coordinates of the dividing point P with the section formula, using m₁ = k and m₂ = 1.
3
Set the coordinate you know equal to its value (for an axis, set y = 0 or x = 0).
4
Solve for k and write the ratio k : 1 in whole numbers.
5
If asked, put k back to find the other coordinate of P.

Example 1: the ratio from a known point

Problem

In what ratio does P(2, 1) divide the segment joining A(−1, 4) and B(6, −3)?

Working

Let the ratio be k : 1. By the section formula, x = (k × 6 + 1 × (−1))/(k + 1)
Set x = 2: (6k − 1)/(k + 1) = 2
6k − 1 = 2k + 2
4k = 3, so k = 3/4
Ratio = 3/4 : 1 = 3 : 4
Check with y: (−3 × 3/4 + 4)/(3/4 + 1) = (7/4)/(7/4) = 1. This matches P.

Answer

P divides AB internally in the ratio 3 : 4.

Example 2: the ratio in which an axis divides

Problem

Find the ratio in which the x-axis divides the segment joining A(3, −4) and B(−2, 6). Also find the point of division.

Working

On the x-axis, y = 0. Let the ratio be k : 1.
y = (k × 6 + 1 × (−4))/(k + 1) = 0
6k − 4 = 0, so k = 2/3. The ratio is 2 : 3.
x = (2 × (−2) + 3 × 3)/(2 + 3) = (−4 + 9)/5 = 1

Answer

The ratio is 2 : 3 and the point of division is (1, 0).

Example 3: an unknown coordinate

Problem

The point P(p, 3) lies on the segment joining A(−4, 9) and B(8, −3). Find the ratio in which P divides AB, and find p.

Working

Let the ratio be k : 1. Use the y-coordinate: (k × (−3) + 1 × 9)/(k + 1) = 3
−3k + 9 = 3k + 3
6 = 6k, so k = 1. The ratio is 1 : 1, so P is the mid-point.
p = (−4 + 8)/2 = 2

Answer

The ratio is 1 : 1 and p = 2.

Tips

Use the coordinate that is given as a number to find k. Use the other one to check or to find an unknown.
If k comes out negative or zero, recheck your working. For a point between A and B, k is always positive.
Write the final ratio in whole numbers: k = 3/4 means 3 : 4.
The x-axis is the line y = 0 and the y-axis is the line x = 0. Students often swap them.

In real life

Suppose a straight water pipe joins two tanks and crosses a road along the x-axis of a map. The k : 1 method tells you exactly where it crosses.

Exam angle

2 marks: “Find the ratio in which the y-axis divides the segment joining (−3, 5) and (6, −1).” On the y-axis x = 0: (6k − 3)/(k + 1) = 0, so k = 1/2. The ratio is 1 : 2, and the point is (0, 3).

3 marks: “In what ratio does the point (1, 2) divide the segment joining (−2, 5) and (7, −4)?” x: (7k − 2)/(k + 1) = 1 gives 6k = 3, so k = 1/2. The ratio is 1 : 2. Check y: (−4 + 10)/3 = 2.

Remember it

Unknown ratio? Call it k : 1. One known coordinate gives one equation and one k.