Circle, ellipse, parabola and hyperbola are all slices of a double cone at different angles.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The circle
(x − h)² + (y − k)² = r² has centre (h, k) and radius r. The signs inside are subtracted, so they flip.
The ellipse
Dividing the two squared terms by different numbers stretches the circle into an ellipse.
Telling them apart
Two squared terms added means a circle or ellipse; subtracted means a hyperbola; only one squared term means a parabola.
All four come from slicing a cone
Cut a double cone with a plane. A horizontal cut gives a circle, a slight tilt an ellipse, a cut parallel to the side a parabola, and a steep cut through both halves a hyperbola. One family, four members.
The circle equation is the distance formula
(x − h)² + (y − k)² = r² says every point is distance r from the centre. Squaring both sides of the distance formula produces it directly, which is why the equation has that shape.
An ellipse has two foci
An ellipse is the set of points whose distances to two fixed foci sum to a constant. Its equation x²/a² + y²/b² = 1 has a and b as the semi-axes, and the larger tells you which way it is stretched.
Where conics appear
Planetary orbits are ellipses, projectile paths parabolas, satellite dishes parabolic, and some comet paths hyperbolic. Conics describe motion under an inverse-square force, which is why they dominate astronomy.
Step 2: Try It Yourself
The circle, and what its radius controls.
- Diameter8 cm
- Circumference25.13 cm
- Area50.27 cm²
- Circumference ÷ diameter3.14
Circumference is a length, measured in cm. Area is a coverage, measured in cm². The units tell them apart.
Step 3: Watch an Example
One step at a time.
Watch Yusuf Read a Circle
Yusuf reads (x − 3)² + (y + 2)² = 25.
- Step 1
The h is 3, because the equation subtracts it.
Step 4: Your Turn
Practice makes it stick.
The Radius
Problem 1 of 2
(x − 1)² + (y − 4)² = 49. What is the radius?
The Centre
Problem 2 of 2
Same circle. What is the x of the centre?
Read the Conic
1 of 8
(x − 2)² + (y − 5)² = 9. Radius?
2 of 8
Same circle. y of the centre?
3 of 8
(x + 4)² + y² = 16. x of the centre?
4 of 8
Match each equation form to the conic it describes.
Tap a card on the left to start.
5 of 8
x² + y² = 100. Radius?
6 of 8
How many conic sections come from slicing a double cone?
7 of 8
Sort each equation by its conic.
Tap something to move it.
- Empty
- Empty
- Empty
8 of 8
(x − 6)² + (y − 6)² = 1. Radius?
Step 5: Quick Check
Show what you know.
Question 1 of 1
(x − 5)² + (y + 1)² = 36. Radius?
What You Learned
- The four conics are slices of a double cone at different angles.
- A circle’s equation gives its centre with the signs flipped, and its radius squared.
- Added squares give a circle or ellipse; subtracted squares give a hyperbola.