P lies on AB extended, with A(4, 1), B(8, −3) and AP : PB = 5 : 1. P = (□, □). Fill in each blank

9, −4. External division: P = ((5 × (8) − 1 × (4))/4, (5 × (−3) − 1 × (1))/4) = (36/4, −16/4) = (9, −4).

A(0, −1), B(−1, 4). P is on the line AB but not between A and B, with AP : PB = 2 : 1. B lies between A and P. True or false?

True. When AP is longer than PB (2 > 1), P is beyond B, so B is between A and P; when AP is shorter, P is beyond A.

A(7, 4), B(0, 5). P is on the line AB but not between A and B, with AP : PB = 1 : 3. P's x-coordinate = □ (as a fraction) Answer in simplest form

21/2. x = (1 × (0) − 3 × (7))/(1 − 3) = −21/−2 = 21/2.

Coordinate geometry · Ages 14–15

External division

External: r x₂ − s x₁r − s.

How to do it

Method

External: (r x₂ − s x₁)/(r − s)

AP : PB = 3 : 1: x = 3 × 3 − 1 × 12 = 4

Another way

Walk it

AB is 2 parts; from B one more: 3 + 2 ÷ 2 = 4

In full

P on AB extended (not between A and B) with AP : PB = r : s divides AB externally. The formula: P = (r x₂ − s x₁r − s, r y₂ − s y₁r − s), like internal division with minus signs for plus. A(1, 2), B(3, 5), AP : PB = 3 : 1: P = (3 × 3 − 1 × 12, 3 × 5 − 1 × 22) = (4, 132). With r > s, P lies beyond B; with r < s, beyond A. Another way: walk it. AP : PB = 3 : 1 makes AB 2 parts and BP 1, so go on from B by half of AB: x by 2 ÷ 2 = 1 and y by 3 ÷ 2 = 32, to (4, 132), the same answer. Check: A, B and P are in one line, and AP and PB are in the ratio r : s.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
P lies on AB extended, with A(4, 1), B(8, −3) and AP : PB = 5 : 1. P = (, ).
Fill in each blank
Show the answer and working
Answer: 9, −4
External division: P = (5 × (8) − 1 × (4)4, 5 × (−3) − 1 × (1)4) = (364, −164) = (9, −4).
Example 2
A(0, −1), B(−1, 4). P is on the line AB but not between A and B, with AP : PB = 2 : 1. B lies between A and P.
True or false?
Show the answer and working
Answer: True
When AP is longer than PB (2 > 1), P is beyond B, so B is between A and P; when AP is shorter, P is beyond A.
Example 3
A(7, 4), B(0, 5). P is on the line AB but not between A and B, with AP : PB = 1 : 3. P's x-coordinate = (as a fraction)
Answer in simplest form
Show the answer and working
Answer: 212
x = 1 × (0) − 3 × (7)1 − 3 = −21−2 = 212.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.A(1, 3), B(4, −6). P is on the line AB but not between A and B, with AP : PB = 1 : 3. B lies between A and P.
True or false?
Answer:
2.P lies on AB extended, with A(−5, −3), B(−11, −9) and AP : PB = 5 : 2. P = (, ).
Fill in each blank
Answer:
3.P lies on BA extended (A between P and B), with A(6, −3), B(12, −9) and AP : PB = 3 : 5. P = (, ).
Fill in each blank
Answer:
4.A(0, 4), B(6, 10). P lies on AB extended with AP : PB = 5 : 3. Which point is P?
- A. (154, 314)
- B. (12, 16)
- C. (−91, −51)
- D. (15, 19)
Choose the right answer
Answer:
5.A(8, −8), B(6, −5). P is on the line AB but not between A and B, with AP : PB = 2 : 5. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:

All the answers

1. False
2. −15, −13
3. −3, 6
4. D. (15, 19)
5. 283

Free worksheet: External divisionWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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