What you'll learn
This topic covers the equation of a circle with its centre at the origin (0, 0) in coordinate geometry, a fundamental concept in CIE IGCSE Mathematics. You'll learn to derive and apply the equation x² + y² = r², determine whether points lie on, inside or outside a circle, and solve problems involving circles and straight lines. This topic regularly appears in Paper 2 and Paper 4 examinations, often combined with other coordinate geometry concepts.
Key terms and definitions
Circle — the set of all points in a plane that are equidistant from a fixed point called the centre.
Radius — the constant distance from the centre of a circle to any point on its circumference, denoted by r where r > 0.
Origin — the point (0, 0) where the x-axis and y-axis intersect on a coordinate plane.
Equation of a circle centred at the origin — the algebraic relationship x² + y² = r² that describes all points (x, y) lying on a circle of radius r with centre at (0, 0).
Circumference — the complete boundary or perimeter of a circle.
Chord — a straight line segment whose endpoints both lie on the circle.
Tangent — a straight line that touches the circle at exactly one point, perpendicular to the radius at that point.
Pythagorean theorem — the mathematical relationship a² + b² = c² in a right-angled triangle, which forms the foundation for deriving the circle equation.
Core concepts
Deriving the equation x² + y² = r²
The equation of a circle centred at the origin stems directly from Pythagoras' theorem. Consider any point P(x, y) on a circle with centre O(0, 0) and radius r.
The distance from O to P must equal r. Using the distance formula or creating a right-angled triangle:
- The horizontal distance from O to P is x
- The vertical distance from O to P is y
- The hypotenuse (the radius) is r
Applying Pythagoras' theorem: x² + y² = r²
This equation holds true for every point on the circle. The standard form x² + y² = r² is the equation you must recognise and use in examinations.
Key observations:
- Both x and y are squared, making the equation symmetrical about both axes
- The coefficient of x² and y² is always 1
- There are no x or y terms (no terms with power 1)
- The right-hand side is always the square of the radius
Finding the radius from the equation
Given an equation in the form x² + y² = k, where k is a positive constant, the radius can be found immediately:
r² = k, therefore r = √k
For example:
- x² + y² = 25 represents a circle with radius 5 (since √25 = 5)
- x² + y² = 12 represents a circle with radius √12 or 2√3
- x² + y² = 0.64 represents a circle with radius 0.8
Important: Always take the positive square root, as radius must be positive. Leave answers in surd form unless the question specifies otherwise or requires a decimal approximation.
Writing the equation given the radius
When given the radius, substitute r² into the standard form.
Process:
- Square the radius value
- Write x² + y² = [squared radius value]
Examples:
- Radius = 7 gives x² + y² = 49
- Radius = 3.5 gives x² + y² = 12.25
- Radius = √10 gives x² + y² = 10
Determining if a point lies on, inside or outside the circle
For a circle with equation x² + y² = r² and a point (a, b):
Substitution method:
- Calculate a² + b²
- Compare with r²
Three cases arise:
- If a² + b² = r², the point lies on the circle
- If a² + b² < r², the point lies inside the circle
- If a² + b² > r², the point lies outside the circle
This is a common exam question requiring clear working. Always show the substitution step explicitly.
Example: For the circle x² + y² = 41, test point (4, -5):
- Substitute: 4² + (-5)² = 16 + 25 = 41
- Since 41 = 41, the point lies on the circle
Example: For the circle x² + y² = 50, test point (3, 6):
- Substitute: 3² + 6² = 9 + 36 = 45
- Since 45 < 50, the point lies inside the circle
Intersection of a circle and a straight line
When a straight line and circle are given, you may need to find their points of intersection. This involves simultaneous equations.
Method:
- Write the equation of the line (usually y = mx + c or x = k or y = k)
- Write the equation of the circle x² + y² = r²
- Substitute the linear equation into the circle equation
- Solve the resulting quadratic equation
- Find corresponding coordinates
The number of intersection points indicates:
- Two distinct solutions: line is a secant (cuts through the circle)
- One solution (repeated root): line is a tangent (touches once)
- No real solutions: line does not intersect the circle
Finding the length of a chord
A chord is a line segment connecting two points on the circle. To find its length:
- Find the coordinates of both endpoints (often by solving simultaneous equations)
- Apply the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Alternatively, if you know the perpendicular distance from the centre to the chord and the radius, use the relationship: (half-chord)² + (perpendicular distance)² = r²
Circle and axes intersections
Circles centred at the origin intersect the coordinate axes at specific points that are straightforward to find.
x-axis intersections (where y = 0): Substitute y = 0 into x² + y² = r² x² = r² x = ±r Points: (r, 0) and (-r, 0)
y-axis intersections (where x = 0): Substitute x = 0 into x² + y² = r² y² = r² y = ±r Points: (0, r) and (0, -r)
The circle always crosses each axis at two points, symmetrically placed about the origin, at distance r from the origin.
Worked examples
Example 1: Basic equation and point testing
Question: A circle has its centre at the origin and passes through the point (5, 12).
(a) Find the equation of the circle. [2 marks] (b) Determine whether the point (8, 9) lies inside, on or outside the circle. [2 marks]
Solution:
(a) First, find the radius using the distance formula or Pythagoras: r² = 5² + 12² = 25 + 144 = 169
The equation is x² + y² = 169
Alternative: r = √169 = 13, so x² + y² = 13² = 169
(b) Substitute (8, 9) into the left side: 8² + 9² = 64 + 81 = 145
Compare: 145 < 169
The point (8, 9) lies inside the circle
Example 2: Intersection with a line
Question: A circle has equation x² + y² = 25. The line y = x - 1 intersects the circle.
(a) Find the coordinates of the points of intersection. [4 marks] (b) Calculate the length of the chord formed. [2 marks]
Solution:
(a) Substitute y = x - 1 into x² + y² = 25:
x² + (x - 1)² = 25
Expand: x² + x² - 2x + 1 = 25
Simplify: 2x² - 2x + 1 = 25
2x² - 2x - 24 = 0
Divide by 2: x² - x - 12 = 0
Factorise: (x - 4)(x + 3) = 0
Therefore: x = 4 or x = -3
When x = 4: y = 4 - 1 = 3, giving point (4, 3)
When x = -3: y = -3 - 1 = -4, giving point (-3, -4)
(b) Distance between (4, 3) and (-3, -4):
d = √[(4 - (-3))² + (3 - (-4))²]
d = √[7² + 7²]
d = √[49 + 49]
d = √98 = 7√2 (or 9.90 to 3 s.f.)
Example 3: Finding radius in surd form
Question: A circle centred at the origin has equation x² + y² = 20.
(a) Write down the exact value of the radius in its simplest form. [2 marks] (b) The point (2, k) lies on the circle, where k > 0. Find the value of k. [2 marks]
Solution:
(a) r² = 20
r = √20 = √(4 × 5) = 2√5
(b) Substitute (2, k) into x² + y² = 20:
2² + k² = 20
4 + k² = 20
k² = 16
k = 4 (since k > 0)
Therefore k = 4
Common mistakes and how to avoid them
• Mistake: Writing the equation as x² + y² = r instead of x² + y² = r² Correction: Always square the radius. The equation contains r-squared, not r. If radius = 6, write x² + y² = 36, not x² + y² = 6.
• Mistake: Forgetting to square negative coordinates when substituting a point Correction: Remember (-5)² = 25, not -25. When testing point (-3, 4), calculate (-3)² + 4² = 9 + 16 = 25, showing all working clearly.
• Mistake: Incorrectly simplifying surds for the radius Correction: If x² + y² = 48, the radius is √48 = √(16 × 3) = 4√3, not 4√12 or left as √48 when simplification is possible.
• Mistake: Stating a point is "not on the circle" rather than specifying inside or outside Correction: When a² + b² ≠ r², you must determine whether the point is inside (a² + b² < r²) or outside (a² + b² > r²). The question specifically asks which.
• Mistake: Finding only one intersection point when solving simultaneous equations Correction: Circle-line intersections typically yield a quadratic equation with two solutions. Find both x-values and their corresponding y-values. Check your factorisation or use the quadratic formula.
• Mistake: Taking the negative square root for radius Correction: Radius is always positive by definition. If x² + y² = 64, then r = 8, never r = -8. Only use the positive root.
Exam technique for "Coordinate Geometry: Equation of a circle centred at the origin"
• Command word "Find the equation" requires you to write x² + y² = [specific number]. You must calculate r² by finding the distance from the origin to a given point, or by squaring a stated radius. Show working for the calculation of r² to earn method marks even if your final answer contains an error.
• Command word "Show that" means the answer is provided—your task is to demonstrate the result through clear algebraic steps. When showing a point lies on a circle, substitute both coordinates, calculate the sum of squares, and explicitly state this equals r². Write a conclusion sentence such as "Since LHS = RHS, the point lies on the circle."
• Simultaneous equations questions typically award 1 mark for correct substitution, 2 marks for solving the quadratic, and 1 mark for finding all coordinate pairs. Set out your work clearly: substitute, expand, collect terms, solve, then find corresponding values. Check both solutions.
• Accuracy requirements: Unless stated otherwise, leave answers involving surds in exact form (e.g., √18 = 3√2). For decimal approximations, use at least 3 significant figures unless specified. Mark schemes penalise premature rounding, so keep full calculator values in intermediate steps.
Quick revision summary
The equation of a circle centred at the origin (0, 0) with radius r is x² + y² = r². To find r from the equation, take the positive square root of the right-hand side. Test whether point (a, b) lies on the circle by checking if a² + b² = r²; if less than r², it's inside; if greater, it's outside. To find intersection points with a line, substitute the line equation into the circle equation and solve the resulting quadratic. Common errors include forgetting to square r and mishandling negative coordinates. This topic combines algebra and coordinate geometry skills essential for IGCSE examination success.