What you'll learn
This topic covers the graphs of circles and other implicit curves. In this guide you will learn the equation of a circle centred on the origin, how to read its radius, how to find where a line meets a circle, how to find the equation of a tangent, and how to recognise other implicit relationships. These are higher-tier graph and algebra skills.
Key terms and definitions
Circle equation — x² + y² = r² for a circle centred on the origin.
Radius — the distance from the centre to the circumference.
Tangent — a line touching the circle at exactly one point.
Implicit equation — an equation linking x and y not written as y = f(x).
Point of intersection — where a line crosses or touches the curve.
Core concepts
Equation of a circle
A circle centred on the origin with radius r has equation x² + y² = r². So x² + y² = 25 is a circle of radius 5. The radius is the square root of the number on the right, and the curve is symmetric about both axes.
Reading the radius
To find the radius from the equation, square root the right-hand side. For x² + y² = 49, the radius is √49 = 7. If the right-hand side is not a perfect square, leave the radius as a surd.
Line meeting a circle
To find where a line meets a circle, substitute the line's equation into the circle equation and solve the resulting quadratic. Two solutions mean the line is a chord (crosses twice), one means it is a tangent (touches once), and none means it misses the circle.
Tangent to a circle
A tangent meets the radius at 90°. At a point on x² + y² = r², the radius has gradient y/x, so the tangent has gradient −x/y (the negative reciprocal). Use this gradient and the point to write the tangent's equation.
Other implicit curves
Some equations link x and y implicitly (not as y = …), such as circles or cubics. Recognise the shape from the form, and solve intersections by substitution as with circles.
Worked examples
Example 1: Radius
State the radius of x² + y² = 36.
√36 = 6.
Example 2: Tangent gradient
Find the gradient of the tangent to x² + y² = 25 at (3, 4).
Radius gradient 4/3, so tangent gradient = −3/4.
Example 3: Intersection count
A line substituted into a circle gives a quadratic with one repeated root. What does this mean?
The line is a tangent — it touches the circle once.
Common mistakes and how to avoid them
Forgetting to square root. The radius is √(right-hand side), not the number itself.
Wrong tangent gradient. It is the negative reciprocal of the radius gradient.
Not substituting fully. Replace y throughout the circle equation.
Miscounting intersections. Use the number of solutions to the quadratic.
Confusing centre. x² + y² = r² is centred on the origin.
Exam technique for Graphs of Circles
Use x² + y² = r² and square root for the radius.
Substitute the line into the circle to find intersections.
Count the solutions to classify the line (chord, tangent or miss).
Use the negative reciprocal of the radius gradient for a tangent.
Recognise the shape of implicit equations.
Quick revision summary
A circle centred on the origin has equation x² + y² = r², so the radius is the square root of the right-hand side (x² + y² = 25 has radius 5). To find where a line meets a circle, substitute the line into the circle equation and solve the quadratic: two solutions mean a chord, one (repeated) means a tangent, none means the line misses. A tangent meets the radius at 90°, so its gradient is the negative reciprocal of the radius gradient (radius gradient y/x gives tangent gradient −x/y); combine with the point to write its equation. Other implicit curves link x and y without a y = form — recognise the shape and solve intersections by substitution. The common errors are forgetting to square root for the radius, the wrong tangent gradient, incomplete substitution, and miscounting intersections. Use x² + y² = r², square root for the radius, substitute lines to find and classify intersections, and use negative reciprocals for tangents.