x² + y² = r² for a circle centred at the origin, points on it, and the tangent at a point.
What you will learn
- Write the equation of a circle centred at the origin
- Decide whether a point is on, inside or outside a circle
- Find the equation of the tangent at a point
x² + y² = r²
Every point (x, y) on a circle with centre (0, 0) and radius r is r away from the origin.
By Pythagoras, x² + y² = r². So x² + y² = 49 has radius √49 = 7 (the number on the right is r², so square root it), and a circle of radius 2.5 is x² + y² = 6.25. A radius of √13 gives x² + y² = 13.
A circle centred at the origin through the point (6, −2) has r² = 6² + (−2)² = 40, so its equation is x² + y² = 40 and its radius is √40 = 6.32.
To find a missing coordinate, substitute the one you know: (5, k) on x² + y² = 169 gives 25 + k² = 169, so k² = 144 and k = 12 (or −12). In the same way, the line y = 3 crosses x² + y² = 25 where x² = 25 − 9 = 16, at x = 4 and x = −4, so the chord between them is 8 long.
To test a point, work out x² + y². Equal to r²: on the circle. Less: inside. More: outside.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- x² + y² = r²
- The tangent at a point
- Try it: See the tangent
7 practice questions, marked as you go; about 25 minutes in all.
In this unit: Circles and circle theorems
The parts of a circle, the circle theorems and how to give them as reasons, tangents, chains of reasoning, and the equation of a circle centred at the origin with its tangents.
- Parts of a circle
- The circle theorems
- Tangents
- Chains of reasoning
- The equation of a circle