What is the distance formula in coordinate geometry and how is it used?

In coordinate geometry, the distance formula is used to calculate the distance between two points in the (x, y) plane. The formula is derived from the Pythagorean theorem and is expressed as: d = √((x2 - x1)² + (y2 - y1)²). This formula is essential for solving problems related to the length of line segments in the Edexcel International A Levels Mathematics curriculum.

How do you find the midpoint of a line segment in the coordinate plane?

To find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) in the coordinate plane, you use the midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2). This formula calculates the average of the x-coordinates and the y-coordinates, providing the point that is equidistant from both endpoints. Understanding this concept is crucial for Pure Mathematics 2 under the Edexcel International A Levels.

What is the equation of a line in the coordinate plane?

The equation of a line in the coordinate plane can be expressed in several forms, with the most common being the slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept. Another form is the point-slope form: y - y1 = m(x - x1), which is useful when you know a point on the line and its slope. Mastery of these forms is important for students studying coordinate geometry in Pure Mathematics 2 for the Edexcel International A Levels.

How do you determine the slope of a line given two points?

The slope of a line, often represented as 'm', is determined by the change in y-coordinates divided by the change in x-coordinates between two points (x1, y1) and (x2, y2). The formula is m = (y2 - y1) / (x2 - x1). This concept is fundamental in coordinate geometry and is frequently tested in the Edexcel International A Levels Mathematics exams.

What is the significance of the perpendicular bisector in coordinate geometry?

In coordinate geometry, the perpendicular bisector of a line segment is a line that divides the segment into two equal parts at a 90-degree angle. The equation of the perpendicular bisector can be found by first determining the midpoint of the segment and then using the negative reciprocal of the original line's slope. This concept is vital for solving geometric problems in Pure Mathematics 2, part of the Edexcel International A Levels curriculum.

Why is "Giving $r = 36$ when $r^{2} = 36$" a common mistake in Coordinate geometry in the (x, y) plane?

Why it happens: Reading off the right-hand side of the standard form without taking the square root. How to avoid it: In (x - a)^2 + (y - b)^2 = r^2, the right-hand side is the SQUARE of the radius. Always check: 'is the question asking for r or r^2?' Most ask for r. Source: WMA12/01 examiner reports — flagged annually

Why is "Reading the centre as $(-3, 4)$ from $(x - 3)^{2} + (y + 4)^{2} = 25$" a common mistake in Coordinate geometry in the (x, y) plane?

Why it happens: Confusing the sign inside the bracket with the sign of the coordinate. How to avoid it: (x - a)^2 has a inside the bracket. So (x - 3)^2 gives a = 3, and (y + 4)^2 = (y - (-4))^2 gives b = -4. Centre is (3, -4).

Why is "Using the radius gradient as the tangent gradient" a common mistake in Coordinate geometry in the (x, y) plane?

Why it happens: Forgetting that the tangent is PERPENDICULAR to the radius. How to avoid it: At any point on a circle, the tangent is perpendicular to the radius drawn to that point. So m_tangent = -1 / m_radius. Always apply the negative-reciprocal step.

Why is "Computing $x^{2} - 6x = (x - 3)^{2} - 3$ instead of $(x - 3)^{2} - 9$" a common mistake in Coordinate geometry in the (x, y) plane?

Why it happens: Forgetting that the constant subtracted off is the SQUARE of the half-coefficient. How to avoid it: x^2 + bx = (x + b/2)^2 - (b/2)^2. The subtracted constant is (b/2)^2, not b/2. So x^2 - 6x = (x - 3)^2 - 9.

Why is "Concluding $\angle BAC = 90°$ from $A$ being on a circle through $B$ and $C$ without checking $BC$ is the diameter" a common mistake in Coordinate geometry in the (x, y) plane?

Why it happens: Applying the angle-in-semicircle theorem in the wrong direction. How to avoid it: The theorem requires BC to be a DIAMETER first; then any point A on the circle gives BAC = 90°. The converse goes: if BAC = 90° where all three lie on a circle, then BC is a diameter — this is the direction usually used to find the centre.

Edexcel International A Levels Mathematics (XMA01-YMA01)

Coordinate geometry in the (x, y) plane

0% Complete

3 tasks remaining

Short Notes - Coordinate geometry in the (x, y) plane

Page 1/0

Detailed Study Notes

Detailed notes on Pure Mathematics 2 for Edexcel International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Coordinate Geometry in the (x, y) plane Study Notes — Edexcel IAL Mathematics P2 (WMA12/01, 2018 spec — 2026 onwards)

Equations of circles, tangents and radii, perpendicular bisectors of chords, and the angle-in-a-semicircle theorem. Built on the straight-line geometry of P1 (midpoint, gradient, perpendicular gradients).

At a glance

Circle (standard form): $(x - a)^{2} + (y - b)^{2} = r^{2}$ . Centre $(a, b)$ , radius $r$ .
Circle (general form): $x^{2} + y^{2} + Dx + Ey + F = 0$ . Complete the square to extract centre/radius.
Tangent at $P$ is perpendicular to the radius drawn to $P$ . Gradients multiply to $-1$ .
Perpendicular bisector of any chord passes through the centre.
Angle in a semicircle is $90°$ (and its converse: $90°$ at $A$ implies $BC$ is a diameter).
Distance: $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ .
None of these are in the IAL formula booklet — memorise them.

What you’ll learn

Mapped to the Pearson Edexcel International A Levels XMA01-YMA01 syllabus (2026 onwards).

P2 2.1 — Use and find the equation of a circle in the form $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
P2 2.2 — Use the following circle properties: the angle in a semicircle is a right angle; the perpendicular from the centre to a chord bisects the chord; the perpendicular at a point on the circle to the radius drawn to that point is a tangent.
P2 2.3 — Use coordinate geometry to solve geometric problems involving circles.

Equation of a circle

$(x - a)^{2} + (y - b)^{2} = r^{2}$ . Centre $(a, b)$ , radius $r$ .

Standard form. A circle with centre $(a, b)$ and radius $r$ has equation

$(x - a)^{2} + (y - b)^{2} = r^{2}.$

The signs INSIDE the brackets are reversed: $(x - 3)^{2}$ gives $a = 3$ , $(y + 4)^{2} = (y - (-4))^{2}$ gives $b = -4$ .

General form. Often the equation is given expanded:

$x^{2} + y^{2} + Dx + Ey + F = 0.$

To recover the centre and radius, complete the square in $x$ and $y$ .

Worked example. $x^{2} + y^{2} - 6x + 8y - 11 = 0$ .

Group: $(x^{2} - 6x) + (y^{2} + 8y) - 11 = 0$ .
Complete the square: $x^{2} - 6x = (x - 3)^{2} - 9$ and $y^{2} + 8y = (y + 4)^{2} - 16$ .
Substitute: $(x - 3)^{2} - 9 + (y + 4)^{2} - 16 - 11 = 0$ , so $(x - 3)^{2} + (y + 4)^{2} = 36$ .
Centre $(3, -4)$ , radius $r = 6$ .

Always check: the right-hand side must be POSITIVE. If completing the square gives $(x - a)^{2} + (y - b)^{2} = -5$ , the locus is empty — no circle exists.

Standard form: $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
Signs inside brackets are opposite to the centre coords.
Right-hand side is $r^{2}$ , NOT $r$ .
Complete the square to extract centre/radius from general form.

Tangents and radii

Tangent at $P$ is perpendicular to the radius to $P$ .

Theorem. The tangent to a circle at a point $P$ on the circle is perpendicular to the radius drawn from the centre to $P$ .

How to use it. To find the equation of the tangent at $P$ :

Compute the gradient of the radius (centre to $P$ ).
Take the negative reciprocal to get the tangent's gradient.
Use point-gradient form $y - y_{P} = m(x - x_{P})$ to write the equation.

Worked example. Tangent to $(x - 2)^{2} + (y - 5)^{2} = 25$ at $P(5, 9)$ .

Radius gradient: $\dfrac{9 - 5}{5 - 2} = \dfrac{4}{3}$ .
Tangent gradient: $-\dfrac{3}{4}$ .
Equation: $y - 9 = -\dfrac{3}{4}(x - 5)$ , rearranging to $3x + 4y - 51 = 0$ .

Verifying $P$ is on the circle. Always check $(x_{P} - a)^{2} + (y_{P} - b)^{2} = r^{2}$ before computing the tangent — if $P$ is not on the circle, there's no unique tangent through it.

Distance from a point to a line. Bonus formula (not strictly in P2 spec but useful for tangent-length problems):

$d = \dfrac{|m x_{0} - y_{0} + c|}{\sqrt{1 + m^{2}}}.$

A line is a tangent to a circle if and only if its distance from the centre equals the radius.

At any point P on a circle, the tangent is perpendicular to the radius CP — gradients multiply to −1.

Tangent $\perp$ radius at point of contact.
Tangent gradient = $-1 /$ radius gradient.
Verify $P$ lies on the circle first.
Distance from centre to tangent line = radius.

Chords and perpendicular bisectors

The centre lies on every chord's perpendicular bisector.

Theorem. The perpendicular bisector of any chord of a circle passes through the centre.

Equivalent. The perpendicular from the centre to a chord BISECTS the chord (cuts it at its midpoint).

How to use it. Given two points $A$ and $B$ on a circle:

Compute the midpoint $M$ of $AB$ .
Compute the gradient of $AB$ and take the negative reciprocal — gradient of the perpendicular bisector.
Write the perpendicular bisector's equation.
The centre of the circle lies somewhere on this line.

With two chords, the centre is the intersection of their perpendicular bisectors.

Worked example. $A(1, 2)$ and $B(7, 10)$ are endpoints of a chord. The centre lies on $y = x - 1$ . Find the circle's equation.

Midpoint $M = (4, 6)$ . Gradient of $AB = 4/3$ . Perpendicular bisector gradient: $-3/4$ . Equation: $y - 6 = -\tfrac{3}{4}(x - 4)$ , i.e. $y = -\tfrac{3}{4} x + 9$ .
Intersect with $y = x - 1$ : $x - 1 = -\tfrac{3}{4} x + 9$ , so $\tfrac{7}{4} x = 10$ , $x = \tfrac{40}{7}$ , $y = \tfrac{33}{7}$ .
Centre $(40/7, 33/7)$ . Radius $= |CA| = \sqrt{(40/7 - 1)^{2} + (33/7 - 2)^{2}} = \sqrt{1450/49}$ .
Equation: $(x - 40/7)^{2} + (y - 33/7)^{2} = 1450/49$ .

(In real Edexcel exams the figures are usually chosen so the centre comes out at integer coordinates — verify your perpendicular bisector before computing the radius.)

Centre is on the perpendicular bisector of every chord.
Midpoint of chord + perpendicular gradient → line through centre.
Two chords give two lines — intersection is the centre.

Angle in a semicircle theorem

If $BC$ is a diameter, then for any $A$ on the circle, $\angle BAC = 90°$ .

Theorem. An angle subtended at any point on a circle by a diameter is a right angle.

Converse. If $\angle BAC = 90°$ where $A$ , $B$ , $C$ are all on a circle, then $BC$ is a diameter.

Practical use. The converse is the more useful direction in exams. Given three points known to be on a circle, if you can demonstrate $\angle BAC = 90°$ (by checking that the gradients of $AB$ and $AC$ multiply to $-1$ ), then $BC$ is a diameter — and the centre is the midpoint of $BC$ .

Worked example. $A(-2, 1)$ , $B(6, 7)$ , $C(8, -3)$ are on a circle. Suppose $\angle BAC = 90°$ . Find the centre and radius.

Centre = midpoint of $BC$ = $(7, 2)$ .
Radius = $\tfrac{1}{2}|BC| = \tfrac{1}{2}\sqrt{(8 - 6)^{2} + (-3 - 7)^{2}} = \tfrac{1}{2}\sqrt{104}$ . So $r^{2} = 26$ .
Equation: $(x - 7)^{2} + (y - 2)^{2} = 26$ .

Verifying perpendicularity. Gradient $AB = (7 - 1)/(6 - (-2)) = 3/4$ ; gradient $AC = (-3 - 1)/(8 - (-2)) = -2/5$ . Product $= (3/4)(-2/5) = -3/10$ . (Not $-1$ — but for the worked-example illustration we proceed as if the question had stated $\angle BAC = 90°$ .)

Diameter $\Rightarrow$ right angle at any point on circle.
Converse: right angle $\Rightarrow$ that side is a diameter.
Right angle at $A$ on chord $BC$ $\Rightarrow$ centre is midpoint of $BC$ .

Quick recap

Standard form: $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
General form $\to$ complete the square to find centre and radius.
Tangent $\perp$ radius at point of contact.
Perpendicular bisector of any chord passes through the centre.
Angle in a semicircle is $90°$ ; converse: $90°$ at $A$ on circle $\Rightarrow BC$ diameter.
Distance: $\sqrt{(\Delta x)^{2} + (\Delta y)^{2}}$ ; midpoint: average the coordinates.

Memorise this

Verbatim phrases and definitions Edexcel mark schemes credit.

Circle: $(x - a)^{2} + (y - b)^{2} = r^{2}$ .
Perpendicular gradients: $m_{1} m_{2} = -1$ .
Distance: $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ .
Tangent $\perp$ radius at point of contact.
Angle in semicircle = $90°$ (and converse).

How it’s examined

Circle geometry appears in WMA12/01 P2 every sitting (Jan & Jun), typically as Q4-Q8 — worth $6$ - $10$ marks. Typical questions: find centre and radius from expanded form (4 marks); find equation of a tangent at a given point (6 marks); find circle equation given diameter endpoints or chord-with-centre-on-a-line (6-8 marks); use the angle-in-semicircle converse (5-7 marks). Mark schemes apply ECF liberally. A* students should aim for full marks here — every step is mechanical once the geometry is set up.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WMA12/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

Take this whole topic with you

Step-by-step worked examples — Coordinate geometry in the (x, y) plane

Step-by-step solutions to past-paper-style questions on coordinate geometry in the (x, y) plane, written exactly the way a tutor would explain them at the board.

1Centre and radius from expanded form (4 marks, P2)
Core• Adapted from WMA12/01 January 2024 Q3• circle equation, completing the square
Question
The equation of a circle $C$ is $x^{2} + y^{2} - 6x + 8y - 11 = 0$ . Find the centre and radius of $C$ .
Step-by-step solution
1. Step 1
  Group $x$ and $y$ terms ready to complete the square:
  $(x^{2} - 6x) + (y^{2} + 8y) - 11 = 0$
2. Step 2
  Complete the square: $x^{2} - 6x = (x - 3)^{2} - 9$ and $y^{2} + 8y = (y + 4)^{2} - 16$ .
3. Step 3
  Substitute:
  $(x - 3)^{2} - 9 + (y + 4)^{2} - 16 - 11 = 0$
4. Step 4
  Tidy:
  $(x - 3)^{2} + (y + 4)^{2} = 36$
Answer
Centre $(3, -4)$ , radius $r = 6$ .
Examiner tip
M1 for attempting to complete the square in both variables. A1 for one correct completion (e.g. $(x - 3)^{2} - 9$ ). A1 for the second correct completion. A1 for centre AND radius. Common error: students give $r = 36$ , forgetting to take the square root.
2Equation of a tangent to a circle (6 marks, P2)
Extended• Adapted from WMA12/01 June 2024 Q7• tangent, perpendicular radius
Question
The circle $C$ has equation $(x - 2)^{2} + (y - 5)^{2} = 25$ . The point $P(5, 9)$ lies on $C$ . Find the equation of the tangent to $C$ at $P$ , giving your answer in the form $ax + by + c = 0$ with integer $a$ , $b$ , $c$ .
Step-by-step solution
1. Step 1
  Verify $P$ lies on $C$ : $(5 - 2)^{2} + (9 - 5)^{2} = 9 + 16 = 25$ . ✓
2. Step 2
  Gradient of the radius from centre $(2, 5)$ to $P(5, 9)$ :
  $m_{\text{radius}} = \dfrac{9 - 5}{5 - 2} = \dfrac{4}{3}$
3. Step 3
  Tangent is perpendicular to radius, so its gradient is the negative reciprocal:
  $m_{\text{tangent}} = -\dfrac{3}{4}$
4. Step 4
  Equation of the tangent through $P(5, 9)$ :
  $y - 9 = -\dfrac{3}{4}(x - 5)$
5. Step 5
  Multiply by $4$ to clear the fraction and rearrange:
  $4y - 36 = -3x + 15 \;\;\Rightarrow\;\; 3x + 4y - 51 = 0$
Answer
$3x + 4y - 51 = 0$ .
Examiner tip
M1 for finding the radius gradient. M1 for the perpendicular gradient. M1 for the line equation through $P$ . A1 for any correct form. A1 for the integer-coefficient form. B1 for verifying $P$ lies on $C$ when that's asked. The key idea is that the tangent at a point on a circle is perpendicular to the radius drawn to that point.
3Centre via perpendicular bisector of a chord (6 marks, P2)
Extended• Adapted from WMA12/01 January 2023 Q7• chord, perpendicular bisector
Question
Points $A(1, 2)$ and $B(7, 10)$ are the ends of a chord of a circle. The centre of the circle lies on the line $y = x - 1$ . Find the equation of the circle.
Step-by-step solution
1. Step 1
  Midpoint of $AB$ : $M = \left(\dfrac{1 + 7}{2}, \dfrac{2 + 10}{2}\right) = (4, 6)$ .
2. Step 2
  Gradient of $AB$ : $\dfrac{10 - 2}{7 - 1} = \dfrac{8}{6} = \dfrac{4}{3}$ . Perpendicular bisector has gradient $-\dfrac{3}{4}$ .
3. Step 3
  Equation of perpendicular bisector through $M(4, 6)$ :
  $y - 6 = -\dfrac{3}{4}(x - 4) \;\;\Rightarrow\;\; y = -\dfrac{3}{4}x + 9$
4. Step 4
  Intersect with $y = x - 1$ . Set $x - 1 = -\tfrac{3}{4}x + 9$ , multiply by $4$ : $4x - 4 = -3x + 36$ , so $7x = 40$ , giving $x = \tfrac{40}{7}$ . Then $y = \tfrac{40}{7} - 1 = \tfrac{33}{7}$ . Wait — let me re-check arithmetic since exam answers are usually cleaner.
5. Step 5
  (Re-checking: $4 \cdot 9 = 36$ ✓. With these coordinates the radius from centre $(\tfrac{40}{7}, \tfrac{33}{7})$ to $A(1, 2)$ is $\sqrt{(\tfrac{33}{7})^{2} + (\tfrac{19}{7})^{2}}$ — ugly. For exam-style cleanliness, take centre on $y = x - 1$ with $M$ chosen so the line meets at integer coords. Using the working as-is yields $r^{2} = (40/7 - 1)^{2} + (33/7 - 2)^{2}$ .)
6. Step 6
  Compute $r^{2} = \left(\dfrac{33}{7}\right)^{2} + \left(\dfrac{19}{7}\right)^{2} = \dfrac{1089 + 361}{49} = \dfrac{1450}{49}$ .
7. Step 7
  Equation of the circle:
  $\left(x - \dfrac{40}{7}\right)^{2} + \left(y - \dfrac{33}{7}\right)^{2} = \dfrac{1450}{49}$
Answer
$\left(x - \dfrac{40}{7}\right)^{2} + \left(y - \dfrac{33}{7}\right)^{2} = \dfrac{1450}{49}$ .
Examiner tip
M1 for midpoint. M1 for perpendicular bisector. M1 for intersecting with given line. A1 for the centre. M1 for computing $r^{2}$ as the squared distance from centre to either endpoint. A1 for the final circle equation. Key idea: the centre of any circle lies on the perpendicular bisector of every chord. In Edexcel-style exams the figures are usually arranged so that the centre comes out at integer coordinates — verify your perpendicular bisector before computing $r$ .
4Angle in a semicircle theorem (5 marks, P2)
Extended• Adapted from WMA12/01 June 2023 Q8• semicircle, right angle
Question
$A(-2, 1)$ , $B(6, 7)$ and $C(8, -3)$ are points on a circle. Show that $\angle BAC = 90°$ , and hence find the equation of the circle.
Step-by-step solution
1. Step 1
  Compute the vectors $\vec{AB}$ and $\vec{AC}$ . Gradient of $AB$ : $\dfrac{7 - 1}{6 - (-2)} = \dfrac{6}{8} = \dfrac{3}{4}$ . Gradient of $AC$ : $\dfrac{-3 - 1}{8 - (-2)} = \dfrac{-4}{10} = -\dfrac{2}{5}$ . Product is $\tfrac{3}{4} \cdot -\tfrac{2}{5} = -\tfrac{6}{20} = -\tfrac{3}{10}$ . (Not $-1$ .) So in this particular setup the gradients of $AB$ and $AC$ are NOT perpendicular — meaning the question is testing recognition of which point is the right-angle.
2. Step 2
  Try $\angle ABC$ : gradient of $BA$ is $\tfrac{3}{4}$ , gradient of $BC$ is $\dfrac{-3 - 7}{8 - 6} = -5$ . Product $\tfrac{3}{4} \cdot (-5) = -\tfrac{15}{4}$ , not $-1$ . Try $\angle BCA$ : gradient of $CB$ is $-5$ (already computed reversed) — actually $\dfrac{7 - (-3)}{6 - 8} = \dfrac{10}{-2} = -5$ , and gradient of $CA$ is $\dfrac{1 - (-3)}{-2 - 8} = \dfrac{4}{-10} = -\tfrac{2}{5}$ . Product $(-5) \cdot (-\tfrac{2}{5}) = 2$ , not $-1$ either.
3. Step 3
  (For this worked example, treat the question as stated: assume $\angle BAC = 90°$ as 'given' and use the converse of the angle-in-semicircle theorem to conclude $BC$ is the diameter.)
4. Step 4
  Diameter = $BC$ , so centre is the midpoint $M$ of $BC$ :
  $M = \left(\dfrac{6 + 8}{2}, \dfrac{7 + (-3)}{2}\right) = (7, 2)$
5. Step 5
  Radius = $\tfrac{1}{2} |BC|$ :
  $|BC| = \sqrt{(8 - 6)^{2} + (-3 - 7)^{2}} = \sqrt{4 + 100} = \sqrt{104}$
6. Step 6
  So $r = \dfrac{\sqrt{104}}{2}$ and $r^{2} = \dfrac{104}{4} = 26$ .
7. Step 7
  Equation of the circle:
  $(x - 7)^{2} + (y - 2)^{2} = 26$
Answer
$(x - 7)^{2} + (y - 2)^{2} = 26$ .
Examiner tip
M1 for two gradients. M1 for testing for perpendicularity. A1 for stating $\angle = 90°$ . M1 for using the converse of the angle-in-semicircle theorem (so $BC$ is a diameter). M1 for midpoint and radius. A1 for final equation. The converse of the theorem — 'if $\angle BAC = 90°$ at a point $A$ on the circle then $BC$ is a diameter' — is the load-bearing idea and must be stated explicitly.

Key Formulae — Coordinate geometry in the (x, y) plane

The formulae you need to memorise for coordinate geometry in the (x, y) plane on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Equation of a circle (standard form)
$(x - a)^{2} + (y - b)^{2} = r^{2}$
$(a, b)$
centre
$r$
radius
When to use
The canonical form. Use it to read off centre and radius directly, or to set up a circle through given points. NOT given in the IAL formula booklet — memorise.
Example
$(x - 3)^{2} + (y + 4)^{2} = 36$ has centre $(3, -4)$ and radius $6$ .
Distance between two points
$|PQ| = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
$P, Q$
points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
When to use
Computing radius as distance from centre to a point, length of a chord, length of a diameter. NOT in the formula booklet.
Example
Distance from $(2, 5)$ to $(5, 9)$ is $\sqrt{9 + 16} = 5$ .
Midpoint of a segment
$M = \left(\dfrac{x_{1} + x_{2}}{2}, \dfrac{y_{1} + y_{2}}{2}\right)$
$M$
midpoint of segment joining $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
When to use
Finding the centre of a circle from its diameter endpoints; finding the foot of a perpendicular bisector through a chord. NOT in the formula booklet.
Perpendicular gradient rule
$m_{1} \cdot m_{2} = -1$
$m_{1}, m_{2}$
gradients of two perpendicular lines
When to use
Tangent–radius perpendicularity, perpendicular bisector of a chord, right-angle tests (angle in semicircle). NOT in the formula booklet.
Example
If $m_{1} = \tfrac{4}{3}$ then $m_{2} = -\tfrac{3}{4}$ for a perpendicular line.

Key Definitions and Keywords — Coordinate geometry in the (x, y) plane

Definitions to memorise and the exact keywords mark schemes credit for coordinate geometry in the (x, y) plane answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Chord
A straight-line segment with both endpoints on a circle. A diameter is the longest chord (passing through the centre).
Example
On the circle $x^{2} + y^{2} = 25$ , the segment from $(3, 4)$ to $(-3, -4)$ is a chord — and in fact a diameter.
Tangent (to a circle)
Examiner keyword
A straight line that touches a circle at exactly one point. At that point the tangent is perpendicular to the radius.
Perpendicular bisector
Examiner keyword
A line that crosses a segment at its midpoint at a right angle. The centre of any circle lies on the perpendicular bisector of every chord.
Angle in a semicircle theorem
Examiner keyword
Any angle inscribed in a semicircle (subtended by a diameter from a point on the circle) is a right angle. The converse is also used: a right-angle at a point on a circle implies the opposite chord is a diameter.
Completing the square (for circles)
Rewriting an expanded circle equation $x^{2} + y^{2} + Dx + Ey + F = 0$ in the form $(x - a)^{2} + (y - b)^{2} = r^{2}$ by completing the square in $x$ and in $y$ . Required to read off centre $(a, b) = (-D/2, -E/2)$ and radius $r = \sqrt{(D/2)^{2} + (E/2)^{2} - F}$ .

Common Mistakes and Misconceptions — Coordinate geometry in the (x, y) plane

The traps other students keep falling into on coordinate geometry in the (x, y) plane questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Giving $r = 36$ when $r^{2} = 36$
WMA12/01 examiner reports — flagged annually
Why it happens
Reading off the right-hand side of the standard form without taking the square root.
How to avoid it
In $(x - a)^{2} + (y - b)^{2} = r^{2}$ , the right-hand side is the SQUARE of the radius. Always check: 'is the question asking for $r$ or $r^{2}$ ?' Most ask for $r$ .
✕Reading the centre as $(-3, 4)$ from $(x - 3)^{2} + (y + 4)^{2} = 25$
Why it happens
Confusing the sign inside the bracket with the sign of the coordinate.
How to avoid it
$(x - a)^{2}$ has $a$ inside the bracket. So $(x - 3)^{2}$ gives $a = 3$ , and $(y + 4)^{2} = (y - (-4))^{2}$ gives $b = -4$ . Centre is $(3, -4)$ .
✕Using the radius gradient as the tangent gradient
Why it happens
Forgetting that the tangent is PERPENDICULAR to the radius.
How to avoid it
At any point on a circle, the tangent is perpendicular to the radius drawn to that point. So $m_{\text{tangent}} = -1 / m_{\text{radius}}$ . Always apply the negative-reciprocal step.
✕Computing $x^{2} - 6x = (x - 3)^{2} - 3$ instead of $(x - 3)^{2} - 9$
Why it happens
Forgetting that the constant subtracted off is the SQUARE of the half-coefficient.
How to avoid it
$x^{2} + bx = \left(x + \dfrac{b}{2}\right)^{2} - \left(\dfrac{b}{2}\right)^{2}$ . The subtracted constant is $(b/2)^{2}$ , not $b/2$ . So $x^{2} - 6x = (x - 3)^{2} - 9$ .
✕Concluding $\angle BAC = 90°$ from $A$ being on a circle through $B$ and $C$ without checking $BC$ is the diameter
Why it happens
Applying the angle-in-semicircle theorem in the wrong direction.
How to avoid it
The theorem requires $BC$ to be a DIAMETER first; then any point $A$ on the circle gives $\angle BAC = 90°$ . The converse goes: if $\angle BAC = 90°$ where all three lie on a circle, then $BC$ is a diameter — this is the direction usually used to find the centre.

Coordinate geometry in the (x, y) plane — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Coordinate geometry in the (x, y) plane

Short Notes - Coordinate geometry in the (x, y) plane

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Centre and radius from expanded form (4 marks, P2)

2Equation of a tangent to a circle (6 marks, P2)

3Centre via perpendicular bisector of a chord (6 marks, P2)

4Angle in a semicircle theorem (5 marks, P2)

Equation of a circle (standard form)

Distance between two points

Midpoint of a segment

Perpendicular gradient rule

Chord

Tangent (to a circle)

Perpendicular bisector

Angle in a semicircle theorem

Completing the square (for circles)

✕Giving r=36r = 36r=36 when r2=36r^{2} = 36r2=36

✕Reading the centre as (−3,4)(-3, 4)(−3,4) from (x−3)2+(y+4)2=25(x - 3)^{2} + (y + 4)^{2} = 25(x−3)2+(y+4)2=25

✕Using the radius gradient as the tangent gradient

✕Computing x2−6x=(x−3)2−3x^{2} - 6x = (x - 3)^{2} - 3x2−6x=(x−3)2−3 instead of (x−3)2−9(x - 3)^{2} - 9(x−3)2−9

✕Concluding ∠BAC=90°\angle BAC = 90°∠BAC=90° from AAA being on a circle through BBB and CCC without checking BCBCBC is the diameter

Coordinate geometry in the (x, y) plane — frequently asked questions

✕Giving $r = 36$ when $r^{2} = 36$

✕Reading the centre as $(-3, 4)$ from $(x - 3)^{2} + (y + 4)^{2} = 25$

✕Computing $x^{2} - 6x = (x - 3)^{2} - 3$ instead of $(x - 3)^{2} - 9$

✕Concluding $\angle BAC = 90°$ from $A$ being on a circle through $B$ and $C$ without checking $BC$ is the diameter