# Circles: the general equation, two circles and loci · Year 12

> Explanation, worked examples and free practice for the national curriculum in England.

In Circles: the general equation, two circles and loci (Year 12) you practise one skill: The general equation of a circle.

Expanding the equation of a circle gives its general form: x squared plus y squared, plus terms in x and in y and a number, all equal to zero. To find the centre and the radius, complete the square: group the terms in x and the terms in y, and add the square of half of each coefficient to both sides. If the right side comes out positive, the figure is a circle and that number is the square of the radius. If it is zero, the figure is a single point, and if it is negative, there are no points at all.

The relative position of two circles depends on the distance between their centres. If it is greater than the sum of the radii, each circle is outside the other. If it equals the sum, the circles touch externally, and if it equals the difference, they touch internally. If it is between the difference and the sum, they intersect at two points, and if it is less than the difference, one circle is inside the other. Subtracting the equations of two intersecting circles cancels the squared terms and leaves the equation of a line: the line that the common chord lies on. When a circle moves, its radius stays the same and its centre moves. A tangent that moves with the circle stays parallel to itself, and the point where it touches moves just as the centre does.

Some loci are circles. Label a general point by its coordinates, write the condition with the distance formula, square, expand and complete the square. For example, the points whose squared distances from two fixed points add up to a fixed number lie on a circle centred at the midpoint between them, if that number is large enough. The points twice as far from one point as from another also lie on a circle.

**Tip:** Before you take a radius from the general equation, check that the number on the right side is positive.

## What you learn

### The general equation of a circle

- Example: The equation x² + y² + 10x − 12y + 57 = 0 describes a circle. What is the x coordinate of its centre?
- Answer: −5
- Solution: Completing the square gives (x + 5)² + (y − 6)² = 4 . So the centre is (−5, 6) , and its x coordinate is −5 .

The exercises are made by rules and the answers are worked out in code. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).

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