Conic sections: circles
The equation of a circle: centre and radius.
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A circle is every point at the same distance from the centre.
Look: (x − 2)² + (y − 2)² = 9: centre (2, 2), radius 3.
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1Circle with centre (0, 0), radius 4?
2Centre of (x + 1)² + y² = 49?
3Radius of x² + y² = 36?
! Careful!
✗Centre of (x − 3)² … is (−3, …).
✓Flip the sign: 3.
✗Radius = 25.
✓25 is r²: radius = 5.
See the whole lesson on one page
A circle is every point at the same distance from the centre.
A circle with centre (h, k) and radius r is written (x − h)² + (y − k)² = r². Read the centre by flipping the signs inside the brackets.
Think of it like thisA goat tied to a pole with a 3 m rope walks in a circle of radius 3 m around the pole.
Solved together
Find the centre and radius of (x − 3)² + (y + 2)² = 25.
Flip the signs inside the brackets for the centre.
centre = (3, −2)The right side is r². Take the square root.
r = √25 = 5
AnswerCentre (3, −2), radius 5.
For older learners and grown-ups: a little more
Expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c); complete the square to get there.
Now practise it
Kanakku makes new questions on conic sections: circles every time. It shows the working and gives a hint when you are stuck.