Study Notes on Conic Sections: Ellipses and Hyperbolas

Definition:
- Conic sections (conics) are the curves obtained by intersecting a cone with a plane.
- Types of Conics: Circle, Parabola, Ellipse, and Hyperbola.
Circle: Produced by cutting the cone with a plane perpendicular to the cone's axis.
Ellipses, Parabolas, Hyperbolas: Result from other angles of intersection.
Goal of the Unit: To find equations whose graphs represent conic sections, with a focus on ellipses and hyperbolas, and to analyze their geometric properties for real-world applications.

On completion of this unit, students will be able to:
- Apply geometric properties of ellipses to sketch their graphs.
- Apply geometric properties of hyperbolas to sketch their graphs.

Definition: An ellipse is an oval curve resembling an elongated circle.
- Mathematical Definition: An ellipse consists of the set of all points (x,y) in a plane such that the sum of their distances from two fixed points, called foci (F1 and F2), is a constant.
Axes of Symmetry: Every ellipse has:
- Two axes of symmetry:
- Major Axis: The longer axis.
- Minor Axis: The shorter axis.
- Vertices and Co-vertices: Each endpoint of the major axis is a vertex, and each endpoint of the minor axis is a co-vertex.
Center: The center of an ellipse is the midpoint of both the major and minor axes, where the axes are perpendicular.
Foci: The foci always lie on the major axis.
Ellipse Orientation: Ellipses can be oriented either vertically or horizontally in the coordinate plane (axes positioned along or parallel to the x- and y-axes).
Cases Considered:
- Ellipses centered at the origin.
- Ellipses centered at points other than the origin.
Key Property: The sum of the distances from any point on an ellipse to the foci equals 2a (the length of the major axis).

Claim: For a point on an ellipse, d1 + d2 = 2a.
- Proof Dynamics:
- If a point is located at the vertex of the ellipse (at (a,0)), then:
  - d1 = a + c (distance to focus F1)
  - d2 = a - c (distance to focus F2)
- Thus, d1 + d2 = (a + c) + (a - c) = 2a.

Relationship between a, b, and c:
- The relationship is defined using the equation: b² = a² - c².
Standard Equation Derivation: Steps involve applying the distance formula.
- Based on the ellipse definition: d1 + d2 = 2a.
- Use distance formulas:
- d1 = $ ext{sqr}(x+c)^2 + y^2$,
- d2 = $ ext{sqr}(x-c)^2 + y^2$
- Simplifying leads to the standard form of an ellipse centered at the origin:
- Final Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ .
Behavior of the Equation: If a > b, the ellipse is stretched horizontally; if b > a, it is stretched vertically.

Standard Form Equations:
- For a horizontal ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where a > b
- For a vertical ellipse: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where a > b
Key Features: Center, vertices, co-vertices, foci, lengths, and positions of axes.
Examples: Horizontal vs Vertical
- Horizontal: Center at origin:
- Vertices: ($\pm a$, 0)
- Co-vertices: (0, $ extpm b$)
- Vertical: Center at origin:
- Vertices: (0, $ extpm a$)
- Co-vertices: ($\pm b$, 0)

Example 1.1: Find the standard form for the ellipse with vertices at ($\pm 8$, 0) and foci at ($\pm 5$, 0).
- Steps to Solution:
- Standard formation yields: $\frac{x^2}{64} + \frac{y^2}{39} = 1$ .
Example 1.2: Find the equation of an ellipse with vertices (0, $ extpm4$) and foci (0, $ extpm\sqrt{15}$).
Example 1.3: Graph the ellipse defined by $\frac{x^2}{9} + \frac{y^2}{25} = 1$ , identifying critical features: center, vertices, co-vertices, foci.

Definition: A hyperbola is defined as the set of all points (x,y) in a plane such that the difference of the distances to two fixed points (foci) is a positive constant.
Axes of Symmetry: Every hyperbola has:
- Two axes of symmetry:
- Transverse Axis: Through the center and vertices.
- Conjugate Axis: Perpendicular to the transverse axis, with endpoints at co-vertices.
Center: The midpoint of both the transverse and conjugate axes.
Foci: Located on the transverse axis.
Asymptotes: Arise from the center and guide the curve's branches as they extend outward.

Here, the derivation comparison to ellipse is akin, leading to:
- Equations:
- $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (horizontal)
- $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (vertical)
Relationship Between a, b, and c: The standard connection is governed by $c^2 = a^2 + b^2$ .

Sketch process involves an initial rectangle structure, followed by drawing asymptotes through the center and plotting the vertices. Graphing branches must adhere to asymptotic behavior, approaching but never crossing.
Example Problems:
- Tasks include finding the vertices, foci, transverse axis lengths, and sketching the complete graph.

Similar principles apply to hyperbolas translated horizontally and vertically:
- Shift gives the standard form for hyperbolas now represented accordingly.

Hyperbola with shifted characteristics will utilize the midpoints and side distances to accurately populate their equations and figures.

Exercise includes sketching, finding equations of ellipses and hyperbolas based on given characteristics, anchoring solutions to learned principles and examples presented in practices.