1. Circles Ellipse Parabolas Hyperbolas
Conic Sections
Circles
Ellipse
Parabolas
Hyperbolas

2. Trick to looking at an equation and knowing what the shape will look like:
Ellipse – you have 2 squared terms that are added together
Circle – you have 2 squared terms with the same coefficient and are added together
Hyperbola – 2 squared terms that are subtracted
Parabola – 1 square term

3. Hyperbolas

4. Hyperbola Formula

5. Example 1 Find the required parts for the hyperbola 𝑦 2 25 − 𝑥 2 9 =1
𝑦 − 𝑥 2 9 =1
Rewrite it with all parts in order
𝑦− − 𝑥− =1
Center:
(0,0)
Foci:
0,± 34
Vertices:
0,5 0,−5
Transverse Axis:
x=0
Asymptotes:
𝑦−0=± 5 3 𝑥−0
𝑦=± 5 3 𝑥
𝑐 2 =25+9
𝑐 2 =34

6. Example 2 Find the required parts for the hyperbola
4 𝑥 2 −9 𝑦 2 +8𝑥+36𝑦=68
4 𝑥 2 +8𝑥−9 𝑦 2 +36𝑦=68
4 (𝑥 2 +2𝑥+1)−9 𝑦 2 −4𝑦+4 =68+4−36
4 𝑥+1 2 −9 𝑦−2 2 =36
𝑥 − 𝑦− =1
Center:
(−1,2)
Foci:
−1± 13 ,2
Vertices:
2,2 −4,2
Transverse Axis:
y=2
Asymptotes:
𝑦−2=± 2 3 𝑥+1
𝑐 2 =9+4
𝑐 2 =13

7. Rectangular Hyperbolas Formula
𝑥𝑦= 𝑘 2 k > 0 Quadrants: 1 and 3 Vertices: (k, k) (-k, -k) k <0 Quadrants: 2 and 4 Vertices: (-k, k) (k, -k)

8. Example 3 𝑥𝑦=25 𝑥𝑦=−4 Quadrants: Quadrants: 1 and 3 2 and 4 Vertices:
(5,5) (-5,-5)
Quadrants:
2 and 4
Vertices:
(-2,2) (2,-2)

9. Practice Problems 𝑦 2 49 − 𝑥 2 36 =1 𝑥+3 2 49 − y−4 2 25 =1 𝑥𝑦=9
𝑦 − 𝑥 =1
𝑥 − y− =1
𝑥𝑦=9
𝑥𝑦=−16