Coordinate geometry in the (x, y) plane

Q: Do I need to write straight line equations in form with a positive leading coefficient?

No, Edexcel only requires integer coefficients for unless the question explicitly states a positive leading coefficient. Either form with integer coefficients is awarded full marks.

Edexcel International A-Level Mathematics· P1 2.1-2.2 (2018 spec)· 25 min read

1. Foundational Coordinate Calculations★★☆☆☆⏱ 5 min

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📘 Definition

Core Coordinate Formulas

For two points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ :
1. Gradient $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
2. Midpoint of $A B$ : $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
3. Distance $A B$ : $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

📐 Worked Example

Points A(2, 3) and B(6, 7) are plotted on the xy plane. Calculate the gradient of AB, the midpoint of AB, and the length of AB, giving your length answer to 2 decimal places.

1
Calculate gradient using the rise over run formula
$m = \frac{7-3}{6-2} = \frac{4}{4} = 1$
2
Calculate midpoint by averaging x and y coordinates
$Midpoint = \left(\frac{2+6}{2}, \frac{3+7}{2}\right) = (4, 5)$
3
Calculate distance using Pythagoras' theorem
$AB = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 5.66 \text{ (2 dp)}$

2. Forms of the Straight Line Equation★★★☆☆⏱ 7 min

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📘 Definition

Straight Line Equation Forms

Three valid forms for Edexcel P1:
1. Slope-intercept: $y = m x + c$ where $m$ is gradient, $c$ is y-intercept
2. Point-slope: $y - y_{1} = m (x - x_{1})$ for line with gradient $m$ passing through $(x_{1}, y_{1})$
3. Standard form: $a x + b y + c = 0$ where $a, b, c$ are integers

📐 Worked Example

Find the equation of the line with gradient 3 passing through point (2, 5), giving your answer in standard form $a x + b y + c = 0$ with integer coefficients.

1
Use point-slope form first as you have a known point and gradient
$y - 5 = 3(x - 2)$
2
Rearrange to slope-intercept form to simplify
$y - 5 = 3x - 6 \\ y = 3x -1$
3
Rearrange all terms to one side to get standard form
$3x - y - 1 = 0$

Exam tip:

If a question does not specify the form of the equation, $y = m x + c$ is acceptable, but standard form is often required for 1 final mark so check question instructions carefully.

3. Parallel and Perpendicular Lines★★★☆☆⏱ 7 min

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📘 Definition

Parallel and Perpendicular Line Rules

For two lines with gradients $m_{1}$ and $m_{2}$ :
1. Parallel lines: $m_{1} = m_{2}$ (equal gradients, different y-intercepts)
2. Perpendicular lines: $m_{1} m_{2} = - 1$ (product of gradients is -1, or one line is horizontal and the other vertical)

📐 Worked Example

Line L1 has equation $y = 2 x + 4$ . Find the equation of line L2 that is perpendicular to L1 and passes through point (4, 1), giving your answer in the form $a x + b y + c = 0$ .

1
Identify gradient of L1 from its equation: $m_{1} = 2$
2
Calculate gradient of perpendicular line L2 using $m_{1} m_{2} = - 1$
$m_2 = -\frac{1}{2}$
3
Use point-slope form for L2 passing through (4,1)
$y - 1 = -\frac{1}{2}(x -4)$
4
Multiply all terms by 2 to eliminate fractions, then rearrange to standard form
$2y - 2 = -x +4 \\ x + 2y -6 = 0$

4. Intersection of Two Lines★★★★☆⏱ 6 min

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To find the intersection point of two straight lines, solve their equations simultaneously using either substitution or elimination. The solution gives the (x,y) coordinates where the lines cross.

📐 Worked Example

Find the intersection point of lines $2 x + y = 7$ and $x - 3 y = 0$ .

1
Rearrange the second equation to isolate x for substitution
$x = 3y$
2
Substitute $x = 3 y$ into the first equation
$2(3y) + y =7 \\ 6y + y =7 \\ 7y =7 \\ y =1$
3
Substitute $y = 1$ back into $x = 3 y$ to find x
$x = 3(1) =3$
4
Verify the solution satisfies both original equations: $2 (3) + 1 = 7$ and $3 - 3 (1) = 0$ , so the intersection point is (3,1)

Exam tip:

Always substitute your calculated coordinates back into both original line equations to check for arithmetic errors, as this takes 10 seconds and avoids losing marks for small mistakes.

5. Common Pitfalls

Wrong move:

Subtracting coordinates in the wrong order when calculating gradient, e.g. $\frac{y _{1} - y _{2}}{x _{2} - x _{1}}$

Why:

This flips the sign of the gradient, leading to incorrect line equations and parallel/perpendicular checks.

Correct move:

Always subtract the coordinates of the first point from the second (or vice versa for both numerator and denominator) to keep the ratio consistent.

Wrong move:

Forgetting to eliminate fractions when writing standard form $a x + b y + c = 0$ , leaving non-integer coefficients.

Why:

Edexcel requires integer coefficients for this form unless stated otherwise, so you will lose 1 mark for non-integer values.

Correct move:

Multiply all terms by the lowest common denominator of all fractions to get integer a, b, c values.

Wrong move:

Using the parallel gradient rule instead of perpendicular, or vice versa.

Why:

This leads to entirely incorrect line equations, costing most or all marks for a question.

Correct move:

Write down $m_{1} = m_{2}$ for parallel and $m_{1} m_{2} = - 1$ for perpendicular at the start of any question asking about line relationships to avoid confusion.

Wrong move:

Assuming that all perpendicular lines follow the $m_{1} m_{2} = - 1$ rule, including horizontal/vertical pairs.

Why:

Horizontal lines have gradient 0, vertical lines have undefined gradient, so their product does not exist, but they are still perpendicular.

Correct move:

Check if either line is horizontal or vertical first before applying the product rule.

Wrong move:

Making arithmetic errors when solving simultaneous equations for line intersections, and not checking the solution.

Why:

Small addition/subtraction errors lead to wrong intersection points, which may be used in later parts of a question, causing cascading mark loss.

Correct move:

Substitute your calculated (x,y) values back into both original line equations to confirm they satisfy both.

6. Quick Reference Cheatsheet

Concept	Formula/Rule	Exam Note
Gradient between 2 points	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	Label points clearly to avoid sign errors
Midpoint of 2 points	$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	Used for line bisector questions
Distance between 2 points	$(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$	Give exact form or rounded as specified
Straight line forms	$y = m x + c$ , $y - y_{1} = m (x - x_{1})$ , $a x + b y + c = 0$	Use point-slope for known point + gradient
Parallel lines	$m_{1} = m_{2}$	Distinct lines have different y-intercepts
Perpendicular lines	$m_{1} m_{2} = - 1$	Exception: horizontal + vertical pairs
Line intersection	Solve simultaneous equations	Check solution in both line equations

7. Frequently Asked

Do I need to write straight line equations in $a x + b y + c = 0$ form with a positive leading coefficient?

No, Edexcel only requires integer coefficients for $a x + b y + c = 0$ unless the question explicitly states a positive leading coefficient. Either form with integer coefficients is awarded full marks.

Is coordinate geometry of circles included in P1?

No, circle coordinate geometry is exclusively part of the P2 (WMA12) unit, and will not appear in P1 (WMA11) exams.

Can I use a calculator for P1 coordinate geometry questions?

Yes, calculators are fully permitted for all P1 exam papers, including coordinate geometry sections, to check gradient or distance calculations.

Going deeper

syllabusEdexcel IAL 2018 Maths SpecificationRefer to section P1 2.1-2.2 for official content requirements

What's Next

Now you have mastered P1 coordinate geometry, you are ready to move to more advanced Pure Maths topics. This foundational knowledge is critical for upcoming units, including coordinate geometry of circles in P2, and applications of differentiation to tangents and normals in P1 and P2. To reinforce your learning, practice as many past paper structured questions as possible, focusing on multi-part problems that combine line equations, perpendicular lines, and intersection calculations. Make sure you can quickly convert between all three forms of straight line equations without arithmetic errors, as this is a common requirement for early parts of exam questions that lead to higher-mark later sections.

→Differentiation (P1)