Algebra 2: Ellipses – Deriving the Equation from the Definition

Understanding the Ellipse Definition 🧐

An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant. Let's derive the equation using this definition.

Setting Up the Problem 📐

• Foci: Let the foci be at $F_1(-c, 0)$ and $F_2(c, 0)$.
• A point on the ellipse: Let $P(x, y)$ be any point on the ellipse.
• Constant sum: Let the constant sum of distances be $2a$ (where $a$ is the semi-major axis).

Applying the Definition ✍️

By definition, the sum of the distances from $P$ to $F_1$ and $F_2$ is $2a$:

$PF_1 + PF_2 = 2a$

Using the distance formula:

$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a$

Isolating and Squaring ➗

1. Isolate one radical: $\sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2}$
2. Square both sides: $(x + c)^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + (x - c)^2 + y^2$

Simplifying the Equation ➕

1. Expand and simplify: $x^2 + 2cx + c^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + x^2 - 2cx + c^2 + y^2$
2. Further simplification: $4cx - 4a^2 = -4a\sqrt{(x - c)^2 + y^2}$
3. Divide by 4: $cx - a^2 = -a\sqrt{(x - c)^2 + y^2}$

Squaring Again and Rearranging 🔄

1. Square both sides again: $(cx - a^2)^2 = a^2((x - c)^2 + y^2)$
2. Expand: $c^2x^2 - 2a^2cx + a^4 = a^2(x^2 - 2cx + c^2 + y^2)$
3. Distribute: $c^2x^2 - 2a^2cx + a^4 = a^2x^2 - 2a^2cx + a^2c^2 + a^2y^2$
4. Rearrange: $a^4 - a^2c^2 = a^2x^2 - c^2x^2 + a^2y^2$
5. Factor: $a^2(a^2 - c^2) = (a^2 - c^2)x^2 + a^2y^2$

Introducing $b^2$ and Finalizing the Equation ✨

Let $b^2 = a^2 - c^2$ (where $b$ is the semi-minor axis). Then:

$a^2b^2 = b^2x^2 + a^2y^2$

Divide by $a^2b^2$:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Result 🎉

The standard equation of an ellipse centered at the origin is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1