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Mom's Basement University

Ellipses and Hyperbolas

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn the standard equations for ellipses and hyperbolas and how to identify their key features.

The Concept

Section titled “The Concept”

Ellipses

Section titled “Ellipses”

An ellipse is an oval-shaped curve. The standard equation with center (h, k) is:

Horizontal major axis (wider than tall, a > b):

(x−h)2a2+(y−k)2b2=1\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

Vertical major axis (taller than wide, a > b):

(x−h)2b2+(y−k)2a2=1\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1

Key features: center (h, k), vertices along the major axis, co-vertices along the minor axis.

Hyperbolas

Section titled “Hyperbolas”

A hyperbola has two separate branches. The standard equations are:

Horizontal transverse axis (branches open left/right):

(x−h)2a2−(y−k)2b2=1\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1

Vertical transverse axis (branches open up/down):

(y−k)2a2−(x−h)2b2=1\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1

Key features: center (h, k), vertices along the transverse axis, asymptotes that the branches approach.

Worked Example

Section titled “Worked Example”

The left graph shows the green ellipse centered at (2, −1) with a = 3 (horizontal) and b = 2 (vertical). The gold dot is the center, and the blue dots are the vertices at (−1, −1) and (5, −1). The right graph shows the purple hyperbola centered at (−3, 5) with two branches opening left and right. The dashed lines are the asymptotes that the branches approach but never touch.

1. Identify the conic and center

(x−2)29+(y+1)24=1\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1

Both terms positive → ellipse. Center: (2, −1). Since 9 > 4, the larger denominator is under x², so a = 3 (horizontal major axis) and b = 2.

2. Identify the conic

(x+3)216−(y−5)29=1\frac{(x + 3)^2}{16} - \frac{(y - 5)^2}{9} = 1

Opposite signs → hyperbola. The positive term is under x², so the transverse axis is horizontal. Center: (−3, 5). a = 4, b = 3.

Real-World Application

Section titled “Real-World Application”
  • Ellipses: planetary orbits (Kepler’s laws), whispering galleries, certain sports stadiums, and gears.
  • Hyperbolas: navigation systems (LORAN), cooling towers at power plants, some telescope designs, and hyperbolic mirrors.

Example: the orbits of planets around the sun are ellipses, with the sun at one focus.

Quiz

Section titled “Quiz”
The equation $\frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1$ is a:
The equation $\frac{(x+3)^2}{25} - \frac{(y-1)^2}{16} = 1$ is a:
In the ellipse $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, if $a > b$, the major axis is:
The center of $\frac{(x+4)^2}{16} - \frac{(y-3)^2}{9} = 1$ is:
For the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$, what is the length of the semi-major axis?