Leaving Certificate Ordinary Level
Coordinate Geometry of the Circle — Notes Summary
Recognise the question type, choose the correct formula, substitute accurately, and show clear working.
Centre (h, k) Radius r On / Inside / Outside Tangent Line–Circle Intersection
1) Key formulas
Most questions start with one of these.
Circle (centre (h, k), radius r)
(x − h)2 + (y − k)2 = r2
Special case: centre (0, 0)
x2 + y2 = r2
Distance between two points
d = √((x2 − x1)2 + (y2 − y1)2)
Quick recognition
  • ▶No brackets in the circle equation ⇒ centre is (0, 0).
  • ▶Brackets present ⇒ centre is (h, k) (read from the brackets).
Circle: Centre & Radius

Recognise the centre (h,k) and radius (r)

Each question shows a circle in the form (x − h)² + (y − k)² = r². Choose the correct centre and radius.

Questions: 5 One at a time Instant feedback
Question 1 of 5
Score: 0 / 0
Circle equation
(x − 2)² + (y + 3)² = 25
What are the centre and radius?
2) Centre & radius / equation
A. Find the centre and radius
  • From (x − h)2 + (y − k)2 = r2: centre (h, k), radius r.
  • From x2 + y2 = r2: centre (0, 0), radius r.
B. Write the equation of a circle
  • Use (x − h)2 + (y − k)2 = r2.
  • Square the radius carefully.
  • Be careful with signs inside brackets.
Tip: If the centre is (h, k), the equation must include (x − h) and (y − k).
3) On / inside / outside
Substitution + comparing two numbers.
Method
Compute:
L = (x − h)2 + (y − k)2
  • ▶Inside if L < r2
  • ▶On if L = r2
  • ▶Outside if L > r2
Centre (0, 0): compute L = x2 + y2 and compare with r2.
4) Radius from a point
A radius is the distance from the centre to a point on the circle.
Centre (0, 0)
r = √(x2 + y2)
Centre (h, k)
r = √((x − h)2 + (y − k)2)
5) Tangents
A tangent touches the circle at exactly one point.
Key fact
radius ⟂ tangent (at the point of contact)
Finding the tangent at P(x₁, y₁)
  1. Find slope of radius CP, where C(h, k):
    mCP = (y₁ − k) / (x₁ − h)
  2. Tangent slope is the negative reciprocal:
    mt = −1 / mCP
  3. Equation through P (point–slope form):
    y − y₁ = mt(x − x₁)
6) Showing a line is tangent
Sometimes you are given a line and asked to show it is tangent.
distance from centre to the line = r
Reminder: Put the line in the form ax + by + c = 0 before using the point-to-line distance formula.
7) Line–circle intersection (centre (0, 0))
In Ordinary Level questions, the line meets the circle in real point(s) — usually two points (sometimes one point if it is a tangent).
Circle: x2 + y2 = r2    and    Line: y = mx + c
Method (simultaneous equations)
  1. Substitute y = mx + c into x2 + y2 = r2.
  2. You get a quadratic in x: Ax2 + Bx + C = 0.
  3. Solve for x.
  4. Substitute each x back into y = mx + c to find y.
  5. Write the intersection points as ordered pairs (x, y).
Short list: knowledge required
  • Equations: (x − h)2 + (y − k)2 = r2 and x2 + y2 = r2.
  • Read centre (h, k) and radius r from the equation.
  • Decide on/inside/outside by comparing L with r2.
  • Use distance (with √) to find a radius from a point.
  • Tangent: radius is perpendicular to tangent; sometimes distance-to-line equals r.
  • Intersection (centre (0,0)): substitute y = mx + c into the circle and solve the quadratic.
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Circle: Centre & Radius

Recognise the centre (h,k) and radius (r)

Each question shows a circle in the form (x − h)² + (y − k)² = r². Choose the correct centre and radius.

Questions: 5 One at a time Instant feedback
Question 1 of 5
Score: 0 / 0
Circle equation
(x − 2)² + (y + 3)² = 25
What are the centre and radius?