Analytic Geometry · Analytic Geometry

The Equation of a Circle Centred at the Origin

2.3 2.4About 70 minutes

Learning goals

I can develop the equation x² + y² = r² using the length formula.
I can find the radius of a circle from its equation, and write the equation from the radius.
I can sketch a circle given its equation in the form x² + y² = r².
I can decide whether a point is on, inside or outside a circle.

Words to know

circle
The set of all points in a plane that are the same distance from a fixed point.

centre
The fixed point that all points on a circle are the same distance from.

radius
The distance from the centre to any point on the circle (plural: radii).

diameter
A chord through the centre; its length is twice the radius.

chord
A segment joining two points on a circle.

symmetry
A shape has symmetry if it matches itself after a reflection or rotation.

Every point the same distance

A cell tower sends a signal 15 km in every direction. A sprinkler waters a lawn out to 4 m. A dog on a 3 m leash, tied to a stake, can reach anything up to 3 m away. In each case the edge of the region is a circle: the set of all points that are the same distance (the radius) from a fixed point (the centre).

You already know how to find the distance between two points. In this lesson you will use the length formula to find the equation of a circle centred at the origin.

Developing the equation

Place the centre at the origin O(0, 0) and let the radius be r. Let P(x, y) be any point on the circle. The distance OP must equal r. By the length formula:

OP = √[(x − 0)² + (y − 0)²] = √(x² + y²)

So √(x² + y²) = r. Square both sides:

x² + y² = r²

This is the equation of a circle with centre (0, 0) and radius r. A point is on the circle exactly when its coordinates make this equation true. It is the Pythagorean relationship again: the legs are x and y and the hypotenuse is the radius.

For any point P(x, y) on the circle, the legs of the right triangle are x and y and the hypotenuse is the radius r. So x² + y² = r² for every point on the circle. (Here P is (4, 3) and r = 5.)

Worked example

Example 1: From radius to equation, and back

(a) Write the equation of the circle with centre (0, 0) and radius 6. (b) Find the radius of x² + y² = 49. (c) Find the radius of x² + y² = 12.

(a) r² = 6² = 36, so the equation is x² + y² = 36.
(b) r² = 49, so r = √49 = 7. (The radius is a length, so we take only the positive root.)
(c) r² = 12, so r = √12 ≈ 3.46.

(a) x² + y² = 36; (b) r = 7; (c) r = √12 ≈ 3.46.

Common trap

The number on the right side is r², not r. The circle x² + y² = 16 has radius 4, not 16. And the circle with radius 9 is x² + y² = 81, not x² + y² = 9.

Worked example

Example 2: Sketch the circle x² + y² = 25

Sketch the graph of x² + y² = 25 and list some points on it.

The centre is (0, 0) and r = √25 = 5.
Intercepts: put y = 0: x² = 25, x = ±5, giving (5, 0) and (−5, 0). Put x = 0: y = ±5, giving (0, 5) and (0, −5).
More points: try x = 3: 9 + y² = 25, y² = 16, y = ±4, giving (3, 4) and (3, −4). By symmetry, (−3, 4), (−3, −4), (4, 3), (4, −3), (−4, 3) and (−4, −3) are also on the circle.
Plot the intercepts and points, then draw a smooth round curve through them (or use a compass set to 5 units).

The circle has centre (0, 0), radius 5, and passes through 12 points with integer coordinates: (±5, 0), (0, ±5), (±3, ±4) and (±4, ±3).

The circle x² + y² = 25. The 12 marked points all have whole-number coordinates; a few are labelled. Each grid square is 1 unit. The circle is symmetric in both axes and in the origin, so if (3, 4) is on it, so are (−3, 4), (3, −4) and (−3, −4).

Worked example

Example 3: Through a given point

A circle centred at the origin passes through (−5, 12). Find its equation and radius.

Substitute the point into x² + y² = r²: (−5)² + 12² = 25 + 144 = 169.
So r² = 169 and r = 13.

x² + y² = 169, radius 13.

Inside, on or outside?

For a point (a, b), compare a² + b² with r²:

a² + b² = r²: the point is on the circle.
a² + b² < r²: the point is closer than r to the centre, so it is inside.
a² + b² > r²: the point is outside.

Worked example

Example 4: Is the farm in range?

A cell tower at the origin covers a circle of radius 15 km. (a) Write the equation of the edge of the coverage area. (b) Is a farm at (9, 11) covered? (c) Is a cabin at (−10, 12) covered?

(a) x² + y² = 15² = 225.
(b) 9² + 11² = 81 + 121 = 202. Since 202 < 225, the farm is inside the circle.
(c) (−10)² + 12² = 100 + 144 = 244. Since 244 > 225, the cabin is outside.

(a) x² + y² = 225. (b) The farm is covered (it is √202 ≈ 14.2 km away). (c) The cabin is not covered (√244 ≈ 15.6 km away).

Worked example

Example 5: A missing coordinate

The point (x, 4) is on the circle x² + y² = 41. Find x.

Substitute y = 4: x² + 16 = 41, so x² = 25.
x = 5 or x = −5. Both work: there are two points on the circle with y = 4.

x = 5 or x = −5, giving the points (5, 4) and (−5, 4).

Just for fun

I'd tell you a joke about circles, but there's no end to it.

(Also, a circle has no corners, so it can never be cornered in an argument.)

Hands-on activity

A string compass on graph paper

You need: graph paper, a pencil, a piece of string or thread about 15 cm long, a ruler, and a pushpin or thumbtack with a piece of cardboard underneath. Grown-up note: pushpins are sharp; a grown-up can hold the pin in place, or you can hold the end of the string down firmly with a finger instead.

Draw x- and y-axes in the middle of the graph paper. Let 1 square = 1 unit.
Tie one end of the string to the pencil. Hold (or pin) the string at the origin so the pencil tip is exactly 5 units away.
Keeping the string tight, draw a full circle.
Check these points: (3, 4), (−4, 3), (0, −5), (2, 4), (4, −4). Which ones does your circle pass through? Use x² + y² to decide whether each point is on, inside or outside the circle.
Draw a second circle with radius √13 ≈ 3.6 units. Which grid points does it pass through? (Hint: 4 + 9 = 13.)

What to look for: (3, 4), (−4, 3) and (0, −5) are on the circle (sum 25). (2, 4) gives 20 < 25: inside. (4, −4) gives 32 > 25: outside. The second circle passes through the 8 points (±2, ±3) and (±3, ±2).

Your turn

Write the equation of the circle with centre (0, 0) and radius 8.

State the radius of x² + y² = 100.

State the exact radius of x² + y² = 20 and round it to two decimal places.

Find the equation of the circle centred at the origin that passes through (6, −8).

Is (5, 6) on, inside or outside x² + y² = 64?

Find the x-intercepts and y-intercepts of x² + y² = 81.

Find the possible values of y if (−2, y) is on x² + y² = 29.

More practice: warm-up

What is the radius of x² + y² = 9?

Write the equation of a circle centred at the origin with radius 1.

Is (0, 0) inside x² + y² = 4?

Is (6, 8) on x² + y² = 100?

Where is the centre of every circle in this lesson?

More practice: on your own

A circle centred at the origin has a diameter of 18. Write its equation.

A circle centred at the origin passes through (−3, √7). Find its equation.

A rotating lawn sprinkler sits at the origin and waters a circle of radius 4 m. Is a garden gnome at (2.5, −3) getting wet?

Sketch x² + y² = 10. Name four points with integer coordinates on it.

Find the length of the chord of x² + y² = 25 that lies on the line y = 3.

Challenge

Find the points where the line y = x meets the circle x² + y² = 50.

Two circles are centred at the origin: x² + y² = 16 and x² + y² = 49. Describe the region of points between them using inequalities, and find its area.

A Ferris wheel has radius 12 m, centred at the origin of a coordinate grid (in metres). A rider is at a point where x = −7.2. How high above or below the centre could the rider be?

Check your understanding

Quiz

What is the equation of the circle with centre (0, 0) and radius 7?

What is the radius of x² + y² = 36?

Which point is on the circle x² + y² = 25?

A circle centred at the origin passes through (−1, 7). Its equation is:

The point (3, 7) relative to x² + y² = 50 is:

The equation x² + y² = r² comes from:

For parents and teachers

Big idea: a circle is all points at a fixed distance from its centre, so the length formula gives its equation: x² + y² = r² for a circle centred at the origin. This lesson teaches expectations 2.3 (develop the equation of a circle with centre (0, 0) and radius r) and 2.4 (find the radius from the equation, write the equation from the radius, and sketch the circle).

How to help: Ask, "What is the radius?" every time you see a circle equation, and listen for the square root. The string-compass activity shows that the circle and the equation describe the same points.

Common difficulties:

Reading r² as r (thinking x² + y² = 16 has radius 16).
Forgetting the ± when solving x² = 25, and so finding only one point.
Sketches that look like ovals: use a compass or plot the four intercepts and four or more extra points.
This lesson is the foundation for later work with circles and with systems made of a line and a curve.