Section 10.1

The Ellipse

Ellipses are one of the biggest topics on Exam 4, and the techniques here — reading standard form, completing the square, finding foci — carry over directly to hyperbolas. Nail this section and hyperbolas will feel like a remix, not a new song.

⚡ Quick Summary

Think of an ellipse as a squished circle — like an oval or the shape of a running track. Here's a fun way to picture it: stick two thumbtacks into a board (these are the foci), loop a piece of string around them, and pull it tight with a pencil. As you drag the pencil around, you'll trace an ellipse. The total length of string stays the same, which is why the *sum* of the distances from any point on the ellipse to the two foci is always constant.

Standard Form (Horizontal Major Axis)

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \quad a > b

Center (h, k), vertices at (h ± a, k), co-vertices at (h, k ± b)

Standard Form (Vertical Major Axis)

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, \quad a > b

Center (h, k), vertices at (h, k ± a), co-vertices at (h ± b, k)

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Every ellipse has a longer direction (the major axis) and a shorter direction (the minor axis). The number $a$ is always the bigger measurement — it tells you how far the ellipse stretches along its longer side. The number $b$ is the shorter measurement. When you look at the equation, whichever fraction has the bigger denominator — that's $a^2$ , and the variable underneath it tells you whether the ellipse is wider (horizontal) or taller (vertical).

The two foci are special points inside the ellipse, always sitting on the major axis. You find how far they are from the center using $c^2 = a^2 - b^2$ (subtract — not add!). The eccentricity $e = c/a$ tells you how squished the ellipse is: if $e$ is close to 0 it's nearly a circle, and if $e$ is close to 1 it's very long and thin. Earth's orbit around the Sun is an ellipse with $e \approx 0.017$ — almost a perfect circle!

When a problem gives you a messy equation (not already in neat fraction form), you'll need to complete the square to rewrite it. The goal is to get it looking like two fractions that equal 1. Once it's in that standard form, everything — center, $a$ , $b$ , vertices, foci — can be read right off the equation.

Interactive Diagram

Key Formulas

Standard Form (Horizontal Major Axis)

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \quad a > b

Center (h, k), vertices at (h ± a, k), co-vertices at (h, k ± b)

Standard Form (Vertical Major Axis)

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, \quad a > b

Center (h, k), vertices at (h, k ± a), co-vertices at (h ± b, k)

Foci Distance

c^2 = a^2 - b^2

c = distance from center to each focus; foci lie on the major axis

Eccentricity

e = \frac{c}{a}, \quad 0 < e < 1

e near 0 → nearly circular; e near 1 → elongated

Key Takeaways

✓ $a$ is always the larger denominator — whichever variable has $a^2$ underneath determines the major axis direction
✓Foci formula: $c^2 = a^2 - b^2$ (minus for ellipses) — foci lie on the major axis, inside the ellipse
✓Eccentricity $e = c/a$ ranges from $0$ (circle) to just below $1$ (very elongated)
✓When completing the square, factor out leading coefficients first, then balance both sides carefully

⚠️ Common Mistakes

✗Assuming the larger denominator is always under the $x$ -term — $a^2$ is the LARGER denominator regardless of which variable it's under; check both
✗Using $c^2 = a^2 + b^2$ (that's for hyperbolas!) — for ellipses it's $c^2 = a^2 - b^2$ with a minus sign
✗When completing the square, forgetting to multiply the added value by the factored-out coefficient — if you add $4$ inside $9(x^2 + \ldots)$ , you're really adding $9 \times 4 = 36$ to the right side
✗Mixing up vertices and co-vertices — vertices are $a$ units along the MAJOR axis, co-vertices are $b$ units along the minor axis

Next: Section 11.1 Builds on: circles

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10.2 The Hyperbola