Training Conic Sections Ellipses
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Ellipses

24 min Conic Sections

Ellipses

Ellipse

The set of all points whose distances from two fixed points (foci) have a constant sum $2a$.

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

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

Key Relationships

$$c^2 = a^2 - b^2$$

Vertices are $a$ units from center along major axis. Foci are $c$ units from center.

Eccentricity: $e = c/a$, where $0 < e < 1$.

Example 1

Find the foci and vertices: $\frac{x^2}{25} + \frac{y^2}{9} = 1$.

$a^2 = 25$, $b^2 = 9$, $c^2 = 16$, $c = 4$. Horizontal major axis.

Vertices: $(\pm 5, 0)$. Foci: $(\pm 4, 0)$.

Example 2

Find the eccentricity of $\frac{x^2}{100}+\frac{y^2}{64}=1$.

$c = \sqrt{100-64} = 6$. $e = 6/10 = 0.6$.

Example 3

Write the equation of an ellipse centered at origin with vertices $(0,\pm 6)$ and foci $(0,\pm 4)$.

Vertical major axis: $a = 6$, $c = 4$, $b^2 = 36 - 16 = 20$.

$$\frac{x^2}{20} + \frac{y^2}{36} = 1$$

Practice Problems

1. Find vertices and foci: $\frac{x^2}{16}+\frac{y^2}{4}=1$.
2. Find $c$: $a=13$, $b=5$.
3. Write the equation: vertices $(\pm 8,0)$, $b=3$.
4. Eccentricity: $\frac{x^2}{49}+\frac{y^2}{25}=1$.
5. Is the major axis horizontal or vertical: $\frac{x^2}{9}+\frac{y^2}{16}=1$?
6. Find the length of the major and minor axes: $\frac{x^2}{36}+\frac{y^2}{25}=1$.
7. Write the equation: foci $(\pm 3,0)$, vertices $(\pm 5,0)$.
8. Center and $a,b$: $\frac{(x-1)^2}{9}+\frac{(y+2)^2}{4}=1$.
9. An ellipse has $e = 0.8$ and $a = 10$. Find $c$ and $b$.
10. If $a = b$, what shape results?
11. Find the co-vertices: $\frac{x^2}{25}+\frac{y^2}{9}=1$.
12. An orbit is elliptical with $a = 150$ million km and $e = 0.017$. Find $c$.
Show Answer Key

1. Vertices $(\pm 4,0)$, $c=\sqrt{12}=2\sqrt{3}$; foci $(\pm 2\sqrt{3},0)$

2. $c = \sqrt{169-25} = 12$

3. $\frac{x^2}{64}+\frac{y^2}{9}=1$

4. $c = \sqrt{24} = 2\sqrt{6}$; $e = 2\sqrt{6}/7 \approx 0.70$

5. Vertical ($a^2 = 16$ is under $y^2$)

6. Major = $2a = 12$, Minor = $2b = 10$

7. $b^2 = 25-9 = 16$; $\frac{x^2}{25}+\frac{y^2}{16}=1$

8. Center $(1,-2)$, $a = 3$, $b = 2$

9. $c = 8$, $b = \sqrt{100-64} = 6$

10. A circle

11. $(0, \pm 3)$

12. $c = 0.017 \times 150 = 2.55$ million km

Circles & Parabolas Hyperbolas