Ellipse and Hyperbola (lecture notes)

Ellipse and Hyperbola

- the third two-dimensional curve “sum is constant” - the fourth two-dimensional curve.
“The absolute value of differences is constant.”
Is the set of all points (x,y) in a plane whose sum from
two fixed points, called foci, is always constant Is the set of all points (x,y) in a plane whose absolute
value of the difference from two fixed points, called the
The Standard Form of Equation (SFE) of an ellipse whose foci, is always constant.
center is located at the origin is written in two forms:
𝑥2
+
𝑦2
= 1 (For horizontal orientation) The Standard Form of Equation (SFE) of an hyperbola
2 𝑏2
𝑎 whose center is located at the origin is written in two
forms:
𝑦2 𝑥2 𝑥2 𝑦2
+ = 1 (For vertical orientation) − 2 = 1 (For horizontal orientation)
𝑎2 𝑏2 𝑎2 𝑏
𝑦2 𝑥2
- a>b − = 1 (For vertical orientation)
𝑎2 𝑏2




Horizontal
Horizontal 𝑥2 𝑦2
𝑥2 𝑦2 − =1
+ =1 𝑎2 𝑏2
𝑎 2 𝑏2 Foci: (±𝑐, 0)
Foci: (±𝑐, 0) Vertices: (±𝑎, 0)
Vertices: (±𝑎, 0) Co-vertices: (0, ±𝑏)
Co-vertices: (0, ±𝑏) 𝑏
Maj. Axis: 2a // Min. Axis: 2b Asymptotes: 𝑦 = ±(𝑎)𝑥
𝑐 2 = 𝑎2 − 𝑏 2 Trans. Axis: 2a // Conj. Axis: 2b
𝑐 2 = 𝑎2 + 𝑏 2
Vertical
𝑦2 𝑥2 Vertical
+ 2=1 𝑥2 𝑦2
𝑎2 𝑏 𝑎 2
− 𝑏2
=1
Foci: (0, ±𝑐) Foci: (±𝑐, 0)
Vertices: (0, ±𝑎) Vertices: (±𝑎, 0)
Co-vertices: (±𝑏, 0) Co-vertices: (0, ±𝑏)
Maj. Axis: 2a // Min. Axis: 2b 𝑏
𝑐 2 = 𝑎2 − 𝑏 2 Asymptotes: 𝑦 = ±(𝑎)𝑥
Trans. Axis: 2a // Conj. Axis: 2b
𝑐 2 = 𝑎2 + 𝑏 2