Finding the Components of an Ellipse

For the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, find the center, vertices, co-vertices, and foci.

Identify the center: The equation is in standard form with $h = 0$ and $k = 0$ = Center: $(0, 0)$

Identify $a^2$ and $b^2$: $a^2 = 25$ (larger denominator), $b^2 = 9$ = $a = 5$, $b = 3$

Determine orientation: Since $a^2 = 25$ is under $x^2$, the major axis is horizontal = Horizontal ellipse

Find vertices: Vertices are at $(\pm a, 0) = (\pm 5, 0)$ = Vertices: $(-5, 0)$ and $(5, 0)$

Find co-vertices: Co-vertices are at $(0, \pm b) = (0, \pm 3)$ = Co-vertices: $(0, -3)$ and $(0, 3)$

Calculate $c$: $c^2 = a^2 - b^2 = 25 - 9 = 16$, so $c = 4$ = $c = 4$

Find foci: Foci are at $(\pm c, 0) = (\pm 4, 0)$ = Foci: $(-4, 0)$ and $(4, 0)$

Answer: Center: $(0, 0)$, Vertices: $(\pm 5, 0)$, Co-vertices: $(0, \pm 3)$, Foci: $(\pm 4, 0)$

Writing the Equation of an Ellipse

Write the equation of an ellipse centered at $(2, -1)$ with a horizontal major axis of length 10 and minor axis of length 6.

Identify the center: Given: center is $(h, k) = (2, -1)$ = $(h, k) = (2, -1)$

Find $a$ and $b$: Major axis = $2a = 10$, so $a = 5$. Minor axis = $2b = 6$, so $b = 3$ = $a = 5$, $b = 3$

Determine orientation: Horizontal major axis means $a^2$ goes under $(x-h)^2$ = Horizontal ellipse

Write the equation: $\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1$ = $\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1$

Answer: $\frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} = 1$

Vertical Ellipse Analysis

For the ellipse $\frac{(x+3)^2}{4} + \frac{(y-2)^2}{16} = 1$, find all key features.

Identify the center: From $(x+3)^2$ and $(y-2)^2$: $h = -3$, $k = 2$ = Center: $(-3, 2)$

Identify $a^2$ and $b^2$: $a^2 = 16$ (larger), $b^2 = 4$ = $a = 4$, $b = 2$

Determine orientation: $a^2 = 16$ is under $(y-2)^2$, so major axis is vertical = Vertical ellipse

Find vertices: Vertices are at $(h, k \pm a) = (-3, 2 \pm 4)$ = Vertices: $(-3, -2)$ and $(-3, 6)$

Find co-vertices: Co-vertices are at $(h \pm b, k) = (-3 \pm 2, 2)$ = Co-vertices: $(-5, 2)$ and $(-1, 2)$

Calculate $c$: $c^2 = a^2 - b^2 = 16 - 4 = 12$, so $c = \sqrt{12} = 2\sqrt{3}$ = $c = 2\sqrt{3} \approx 3.46$

Find foci: Foci are at $(h, k \pm c) = (-3, 2 \pm 2\sqrt{3})$ = Foci: $(-3, 2 - 2\sqrt{3})$ and $(-3, 2 + 2\sqrt{3})$

Answer: Center: $(-3, 2)$, Vertices: $(-3, -2)$ and $(-3, 6)$, Co-vertices: $(-5, 2)$ and $(-1, 2)$, Foci: $(-3, 2 \pm 2\sqrt{3})$

Mistake: Using $c^2 = a^2 + b^2$ (Pythagorean theorem) instead of $c^2 = a^2 - b^2$

Why: The relationship $c^2 = a^2 + b^2$ is for hyperbolas, not ellipses. For ellipses, the foci are inside the curve, so $c < a$.

Correct: For ellipses: $c^2 = a^2 - b^2$. For hyperbolas: $c^2 = a^2 + b^2$.

Mistake: Confusing which denominator is $a^2$ and which is $b^2$

Why: By convention, $a$ is always the larger value. The position of $a^2$ determines orientation, not which variable it's under.

Correct: Always identify the larger denominator first — that's $a^2$. Its position (under $x$ or $y$ term) tells you the orientation.

Mistake: Forgetting to take the square root when finding $a$, $b$, or $c$

Why: The equation gives $a^2$ and $b^2$, not $a$ and $b$ directly.

Correct: If $a^2 = 25$, then $a = 5$ (not 25). Always square root the denominators to get the actual axis lengths.

Mistake: Mixing up vertices and foci positions for vertical vs horizontal ellipses

Why: Students often place foci along the wrong axis.

Correct: Foci and vertices are ALWAYS on the major axis. If the major axis is horizontal, both are at $(h \pm c, k)$ and $(h \pm a, k)$.

Planetary Orbits

Johannes Kepler discovered that planets orbit the Sun in elliptical paths, with the Sun at one focus.

Earth's orbit has $a \approx 149.6$ million km and eccentricity $e \approx 0.017$, making it nearly circular but technically elliptical.

Whispering Galleries

In an elliptical room, a whisper at one focus can be heard clearly at the other focus due to reflection properties.

The National Statuary Hall in the US Capitol has this property — a whisper on one side can be heard 40 feet away at the opposite focus.

Q: What's the difference between an ellipse and an oval?

A: Mathematically, an ellipse has a precise definition with two foci where the sum of distances from any point to the foci is constant. An oval is a general term for any egg-shaped curve. All ellipses are ovals, but not all ovals are ellipses.

Q: When is an ellipse actually a circle?

A: When $a = b$, the ellipse becomes a circle. In this case, $c = 0$ (since $c^2 = a^2 - b^2 = 0$), meaning the two foci merge into a single point — the center.

Q: What is eccentricity?

A: Eccentricity $e = \frac{c}{a}$ measures how "stretched" an ellipse is. For ellipses, $0 \leq e < 1$. A circle has $e = 0$ (not stretched), while values close to 1 indicate a very elongated ellipse.