Graphs of Circles and Implicit Curves — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the equation of a circle centred on the origin and the curves whose equations link x and y without being written as y = something. By the end of this guide you should be able to read a circle's radius from its equation, write the equation from a radius or from a point on the circle, and test whether a given point lies on, inside or outside it.

You should also be able to find where a straight line meets a circle, find the equation of a tangent at a given point, and recognise the shapes of the other standard curves on the specification.

The organising idea is that x² + y² = r² is Pythagoras' theorem in disguise. Take any point on a circle centred on the origin: its x-coordinate and y-coordinate are the two shorter sides of a right-angled triangle whose hypotenuse is the radius. That is the whole equation. Seeing it this way explains why the radius is the square root of the right-hand side, why (3, 4) lies on x² + y² = 25, and why the numbers in these questions are so often the familiar triples.

Key terms and definitions

Circle equation (centre origin) — x² + y² = r², where r is the radius.

Radius — the distance from the centre to any point on the circle. It is the square root of the number on the right.

Diameter — twice the radius.

Tangent — a straight line touching the circle at exactly one point.

Chord — a straight line cutting the circle at two points.

Implicit equation — an equation relating x and y that is not arranged as y = f(x). The circle equation is the main example at GCSE.

Core concepts

The equation and its radius

A circle centred on the origin with radius r has equation x² + y² = r².

The right-hand side is the radius squared, not the radius. So x² + y² = 25 has radius 5, and x² + y² = 49 has radius 7. This is the most frequently made mistake in the topic: reading x² + y² = 4 as a circle of radius 4 rather than radius 2.

If the right-hand side is not a perfect square, leave the radius as a surd. The circle x² + y² = 13 has radius √13.

Going the other way, a circle of radius 10 has equation x² + y² = 100, because it is the square that appears.

Every such circle is symmetrical about both axes and about the origin, and it crosses each axis at a distance r from the centre — so x² + y² = 9 crosses the positive x-axis at (3, 0) and the positive y-axis at (0, 3).

Testing a point

Substitute the coordinates and compare with r².

Does (6, 8) lie on x² + y² = 100? Substituting gives 36 + 64 = 100, which matches exactly, so yes.

The comparison also tells you more than a yes or no. If the total is less than r², the point is inside the circle; if it is greater, the point is outside. So (1, 2) gives 5, which is less than 25, placing it inside x² + y² = 25.

Writing the equation through a given point

If a circle is centred on the origin and passes through a particular point, substituting that point gives r² directly.

Through (5, 12): 25 + 144 = 169, so the equation is x² + y² = 169 and the radius is 13. The triple 5, 12, 13 appears here for exactly the Pythagorean reason described above.

Where a line meets a circle

Substitute the line's equation into the circle equation and solve the quadratic that results.

The number of solutions answers the geometric question. Two solutions mean the line is a chord, cutting the circle twice. One repeated solution means it is a tangent, touching once. No real solutions mean the line misses the circle entirely.

This is the same "how many roots" reasoning used for quadratic graphs, applied to a new situation.

Tangents

A tangent meets the radius at the point of contact at 90°. That single fact generates the whole method.

The radius from the origin to the point (x, y) has gradient y/x. Perpendicular gradients multiply to −1, so the tangent's gradient is the negative reciprocal, −x/y.

At the point (3, 4) on x² + y² = 25, the radius has gradient 4/3, so the tangent has gradient −3/4. Using that gradient with the point gives the equation of the tangent.

Note the sign. A positive radius gradient always produces a negative tangent gradient, and vice versa, because one must be the negative reciprocal of the other.

Other implicit curves

An implicit equation is simply one not rearranged into y = f(x). Recognising the standard shapes is enough at this level.

y = x³ and other cubics give a curve that rises, levels briefly and rises again, passing through the origin.

y = 1/x, a reciprocal graph, has two separate branches and never touches either axis.

y = kˣ, an exponential graph, rises increasingly steeply and approaches the x-axis without meeting it.

For any of these, intersections with a line are found by the same substitution method used for circles.

Worked examples

Example 1: Reading and using the radius

A circle has equation x² + y² = 36. State its radius and diameter, and give the coordinates where it crosses the positive y-axis.

The right-hand side is the radius squared, so the radius is √36 = 6.

The diameter is twice that: 12.

On the y-axis, x = 0, so y² = 36 and y = 6. The crossing on the positive side is (0, 6).

Example 2: Testing points

Which of the points (3, 4), (5, 1) and (2, 2) lie on the circle x² + y² = 25?

For (3, 4): 9 + 16 = 25. It lies on the circle.

For (5, 1): 25 + 1 = 26, which is greater than 25, so this point is just outside.

For (2, 2): 4 + 4 = 8, which is much less than 25, so this point is inside.

Only the first lies on the circle. Notice how close the second is — a point can miss by a small margin, so the arithmetic must be exact rather than approximate.

Example 3: A tangent

Find the gradient of the tangent to x² + y² = 100 at the point (6, 8).

First check the point is on the circle: 36 + 64 = 100. ✓

The radius to (6, 8) has gradient 8/6, which simplifies to 4/3.

The tangent is perpendicular to the radius, so its gradient is the negative reciprocal: −3/4.

The sign change is the part to watch. Simply inverting to 3/4 would give a line parallel to nothing useful, and would miss the perpendicularity the whole method rests on.

Common mistakes and how to avoid them

Taking the right-hand side as the radius. It is the radius squared. For x² + y² = 4 the radius is 2.

Squaring when you should square root, and vice versa. Going from radius to equation you square; going from equation to radius you take the root.

Forgetting the negative when finding a tangent gradient. Perpendicular means the negative reciprocal, so both the inversion and the sign change are needed.

Assuming a point lies on the circle without checking. Substitute and compare with r² before using the point in any further work.

Losing the second intersection point. A line cutting a circle meets it twice, so a quadratic with two solutions has two answers, not one.

Leaving an awkward radius as a decimal when an exact answer is wanted. √13 is exact; 3.6 is not.

Exam technique for "Graphs of Circles and Implicit Curves"

Write down r before doing anything else. One line — "r² = 25 so r = 5" — makes the rest of the question straightforward and often carries a mark on its own.

Substitute points rather than estimating from a sketch. These questions are decided by exact arithmetic, and a point can lie only one unit outside the circle.

For a tangent, say explicitly that the tangent is perpendicular to the radius. That statement is the method mark, and the arithmetic follows from it.

When a line meets a circle, present the quadratic you obtain and say what its number of solutions means geometrically. The interpretation is usually worth as much as the algebra.

Give coordinates as pairs in brackets and keep exact values as surds where the question asks for an exact answer.

Quick revision summary

A circle centred on the origin has equation x² + y² = r², which is Pythagoras' theorem applied to the coordinates of any point on it.

The number on the right is the radius squared: x² + y² = 49 has radius 7, and a circle of radius 10 has equation x² + y² = 100. The diameter is twice the radius.

Substitute a point to test it: equal to r² means on the circle, less means inside, greater means outside. A circle through (5, 12) has equation x² + y² = 169.

A line meets a circle where the substituted equation's quadratic has solutions — two for a chord, one for a tangent, none for a miss.

A tangent is perpendicular to the radius, so where the radius to (x, y) has gradient y/x, the tangent has gradient −x/y: the negative reciprocal, with both the inversion and the sign change.

Recognise the other standard curves: cubics through the origin, reciprocal graphs in two branches that never touch the axes, and exponential graphs that approach the x-axis without meeting it.