Cogito
Precalculus · Chapter 2 · Lesson 1
Circles and Ellipses
Slicing a cone, and what the equation records.
11 problems · about 24 minutes · TEKS P.3.B, P.3.E
What this lesson teaches
The student identifies a conic from its equation and reads a circle’s centre and radius.
- 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.
Warm Up
Straightforward practice. Get the method working first.
4 problems(x − 5)² + (y + 1)² = 36. Radius?
Answer 6
Why 6.
(x − 2)² + (y − 5)² = 9. Radius?
Answer 3
Why Square root of 9.
Same circle. y of the centre?
Answer 5
Why Subtracted, so it flips.
(x + 4)² + y² = 16. x of the centre?
Answer -4
Why (x + 4) is (x − (−4)).
Build It Up
The same ideas with more to keep track of.
3 problemsMatch each equation form to the conic it describes.
Answer Two squares added, equal coefficients → Circle; Two squares added, different coefficients → Ellipse; Two squares subtracted → Hyperbola; Only one squared term → Parabola
Why The sign between the squares is the first thing to check.
x² + y² = 100. Radius?
Answer 10
Why Centred at the origin.
How many conic sections come from slicing a double cone?
Answer 4
Why Circle, ellipse, parabola, hyperbola.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each equation by its conic.
Answer Circle: x² + y² = 4 · Hyperbola: x² − y² = 4 · Parabola: y = x² + 1
Why Count the squared terms, then check the sign.
(x − 6)² + (y − 6)² = 1. Radius?
Answer 1
Why Square root of 1.
The Radius: (x − 1)² + (y − 4)² = 49. What is the radius?
Answer 7
Why 7.
The Centre: Same circle. What is the x of the centre?
Answer 1
Why 1.