Ellipse - Conic Sections and Properties

1️⃣ Understand the definition of an ellipse and its components

2️⃣ Master ellipse equations in standard and general forms

3️⃣ Identify key elements: center, vertices, co-vertices, foci

4️⃣ Calculate eccentricity and understand c² = a² - b²

5️⃣ Apply ellipse properties to real-world applications

1️⃣ Definition of an Ellipse

An ellipse is the locus of all points where the sum of distances from two fixed points (foci) is constant. It is an oval-shaped conic section that appears in planetary orbits, architectural designs, and optical systems.

$$PF_1 + PF_2 = 2a$$ (constant) for any point P on the ellipse

Key Difference from Hyperbola: For ellipses, $$c^2 = a^2 - b^2$$ (subtraction), while for hyperbolas, $$c^2 = a^2 + b^2$$ (addition)

Center

The midpoint between the two foci

Foci (F₁, F₂)

Two fixed points used in the definition

Vertices

Points on the major axis closest and farthest from center

2️⃣ Ellipse Equations and Forms

a) Horizontal Ellipse (Major axis is horizontal)

Major axis extends horizontally - ellipse is wider than it is tall

Standard Form - Center at Origin

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $$a > b > 0$$

General Form - Center at (h,k)

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

Properties of Horizontal Ellipse:

Vertices: $$(h \pm a, k)$$
Co-vertices: $$(h, k \pm b)$$
Foci: $$(h \pm c, k)$$ where $$c^2 = a^2 - b^2$$
Major axis: horizontal with length $$2a$$
Minor axis: vertical with length $$2b$$

b) Vertical Ellipse (Major axis is vertical)

Major axis extends vertically - ellipse is taller than it is wide

Standard Form - Center at Origin

$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$ where $$a > b > 0$$

General Form - Center at (h,k)

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

Properties of Vertical Ellipse:

Vertices: $$(h, k \pm a)$$
Co-vertices: $$(h \pm b, k)$$
Foci: $$(h, k \pm c)$$ where $$c^2 = a^2 - b^2$$
Major axis: vertical with length $$2a$$
Minor axis: horizontal with length $$2b$$

Key Identification Rules

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

Larger denominator under $$x^2$$
$$a > b$$ ($$a^2$$ under $$x^2$$)
Major axis is horizontal

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

Larger denominator under $$y^2$$
$$a > b$$ ($$a^2$$ under $$y^2$$)
Major axis is vertical

3️⃣ Geometric Relationships

Fundamental Relationship

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

where $$a > b > 0$$ and $$c \geq 0$$

$$a$$

Semi-major axis length

Always the larger value

$$b$$

Semi-minor axis length

Always the smaller value

$$c$$

Distance from center to focus

$$c < a$$ always

4️⃣ Eccentricity

Eccentricity Definition

$$e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$$

Properties of Ellipse Eccentricity:

$$0 \leq e < 1$$ always for ellipses
$$e = 0$$: perfect circle ($$a = b$$)
$$e$$ close to 0: nearly circular
$$e$$ close to 1: very elongated (flattened)
$$e = 1$$: degenerate case (parabola)

5️⃣ Worked Examples

Example 1: Horizontal Ellipse Analysis

Given: $$\frac{x^2}{25} + \frac{y^2}{9} = 1$$

Analysis:

• Center: $$(0, 0)$$
• $$a^2 = 25 \rightarrow a = 5$$ (larger denominator under $$x^2$$)
• $$b^2 = 9 \rightarrow b = 3$$
• $$c = \sqrt{a^2 - b^2} = \sqrt{25 - 9} = 4$$
• Horizontal ellipse ($$a^2$$ under $$x^2$$)
• Vertices: $$(\pm 5, 0)$$
• Co-vertices: $$(0, \pm 3)$$
• Foci: $$(\pm 4, 0)$$
• Eccentricity: $$e = \frac{4}{5} = 0.8$$

Example 2: Vertical Ellipse

Given: $$\frac{(x-2)^2}{4} + \frac{(y+1)^2}{16} = 1$$

Analysis:

• Center: $$(2, -1)$$
• $$a^2 = 16 \rightarrow a = 4$$ (larger denominator under $$y^2$$)
• $$b^2 = 4 \rightarrow b = 2$$
• $$c = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3}$$
• Vertical ellipse ($$a^2$$ under $$y^2$$)
• Vertices: $$(2, -1 \pm 4) = (2, 3)$$ and $$(2, -5)$$
• Co-vertices: $$(2 \pm 2, -1) = (4, -1)$$ and $$(0, -1)$$
• Foci: $$(2, -1 \pm 2\sqrt{3})$$
• Eccentricity: $$e = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \approx 0.87$$

Example 3: Finding Ellipse Equation

Find equation of ellipse with center $$(0,0)$$, vertices $$(\pm 6, 0)$$, and foci $$(\pm 4, 0)$$

Solution:

• Vertices on x-axis → horizontal ellipse
• $$a = 6$$ (distance from center to vertex)
• $$c = 4$$ (distance from center to focus)
• $$b^2 = a^2 - c^2 = 36 - 16 = 20$$
• $$b = \sqrt{20} = 2\sqrt{5}$$
• Equation: $$\frac{x^2}{36} + \frac{y^2}{20} = 1$$

6️⃣ Real-World Applications

Astronomy & Physics

Planetary orbits around the sun
Satellite orbits around planets
Elliptical reflectors and mirrors
Particle accelerator paths

Engineering & Architecture

Elliptical domes and arches
Whispering galleries
Medical ultrasound focusing
Optical lens design

Example: Planetary Orbits

Earth's orbit around the Sun is an ellipse with the Sun at one focus. The orbit has an eccentricity of about 0.017, making it nearly circular. Mars has a more elliptical orbit with eccentricity ≈ 0.093, causing significant variation in its distance from the Sun throughout the year.

7️⃣ Problem-Solving Tips

💡 Strategy for Ellipse Problems

Identify orientation: Compare denominators to find which is larger
Determine $$a$$ and $$b$$: $$a$$ goes with the larger denominator
Find the center: From $$(x-h)^2$$ and $$(y-k)^2$$ terms
Calculate $$c$$: Use $$c^2 = a^2 - b^2$$
Locate vertices and foci: On the major axis
Find eccentricity: $$e = \frac{c}{a}$$ (should be $$< 1$$)

Key Takeaways

1️⃣ Ellipses are defined by constant sum of distances to two foci

2️⃣ Key relationship: $$c^2 = a^2 - b^2$$ (opposite of hyperbola)

3️⃣ Horizontal: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, Vertical: $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$

4️⃣ Always $$a > b > 0$$ and $$0 \leq e < 1$$

5️⃣ Widely used in astronomy, architecture, and optics

Ellipse

All about Ellipses with Simulation