Study Notes: Conic Sections – Circles and Parabolas (Precalculus, Chapter 8.1)

Conic Sections

Overview of Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four main types are circle, ellipse, parabola, and hyperbola. Each has unique geometric properties and algebraic equations.

• Circle: All points equidistant from a fixed center.

• Ellipse: All points where the sum of distances to two foci is constant.

• Parabola: All points equidistant from a fixed point (focus) and a fixed line (directrix).

• Hyperbola: All points where the difference of distances to two foci is constant.

Circles

Definition and Properties

A circle is the set of all points (x, y) in a plane that are an equal distance r from a fixed point (h, k), called the center. The distance r is the radius.

Center–Radius Form of the Equation

• The equation of a circle with center (h, k) and radius r is: $(x - h)^2 + (y - k)^2 = r^2$

• If the center is at the origin (0, 0): $x^2 + y^2 = r^2$

Examples: Finding the Equation of a Circle

• Example 1: Center at (-3, 4), radius 6: $(x + 3)^2 + (y - 4)^2 = 36$ Domain: [-9, 3] Range: [-2, 10]

• Example 2: Center at origin, radius 3: $x^2 + y^2 = 9$ Domain and Range: [-3, 3]

Graphing Circles

• To graph $x^2 + y^2 = 9$: $y = ext{±} ig( ext{sqrt}(9 - x^2) ig)$

• To graph $(x + 3)^2 + (y - 4)^2 = 36$: $y = 4 ext{±} ig( ext{sqrt}(36 - (x + 3)^2) ig)$

General Form of the Equation of a Circle

The general form is: $x^2 + y^2 + Dx + Ey + F = 0$ Depending on the values of D, E, and F, the graph may be a circle, a point, or empty.

Finding the Center and Radius by Completing the Square

• Rewrite the general form to center-radius form by completing the square for both x and y terms.

• Example: $x^2 - 6x + y^2 + 10y + 25 = 0$ Complete the square: $(x - 3)^2 + (y + 5)^2 = 9$ Center: (3, -5), Radius: 3

• If the constant on the right is negative, the equation has no real solutions (no graph).

Parabolas

Definition and Properties

A parabola is the set of all points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix).

Standard Forms of Parabola Equations

• Vertical axis, vertex at (0, 0): $x^2 = 4cy$ Opens upward if $c > 0$, downward if $c < 0$.

• Horizontal axis, vertex at (0, 0): $y^2 = 4cx$ Opens right if $c > 0$, left if $c < 0$.

Examples: Determining Parabola Properties from Equations

• Example 1: $x^2 = 8y$ $4c = 8 ightarrow c = 2$ Focus: (0, 2), Directrix: y = -2, Vertex: (0, 0), Axis: y-axis

• Example 2: $y^2 = -28x$ $4c = -28 ightarrow c = -7$ Focus: (-7, 0), Directrix: x = 7, Vertex: (0, 0), Axis: x-axis

Writing Equations of Parabolas

• Given focus (2/3, 0) and vertex (0, 0): $y^2 = 4 imes rac{2}{3} x = rac{8}{3} x$

• Given vertical axis, vertex (0, 0), through (-2, 12): $x^2 = 4c y$ Substitute: $(-2)^2 = 4c(12) ightarrow 4 = 48c ightarrow c = rac{1}{12}$ $x^2 = rac{1}{3} y$ or $y = 3x^2$

Equation Forms for Translated Parabolas

• Vertex (h, k): $(x - h)^2 = 4c(y - k)$ $(y - k)^2 = 4c(x - h)$

Graphing Parabolas with Vertex (h, k)

• Example: $x = - rac{1}{8}(y + 3)^2 + 2$ Rewrite: $(y + 3)^2 = -8(x - 2)$ Vertex: (2, -3), Focus: (0, -3), Directrix: x = 4

Completing the Square to Graph Parabolas

• Example: $x = -2y^2 + 4y - 3$ Complete the square: $(y - 1)^2 = - rac{1}{2}(x + 1)$ Vertex: (-1, 1), Domain: $(- ext{∞}, -1]$, Range: $(- ext{∞}, ext{∞})$

Application: Parabolic Dish

• Example: Parkes radio telescope, diameter 210 ft, depth 32 ft. Place vertex at origin, opening upward. $x^2 = 4cy$ Passes through (105, 32): $(105)^2 = 4c(32) ightarrow 11,025 = 128c ightarrow c = rac{11,025}{128} ightarrow c ext{ (focus distance)}$ Equation: $x^2 = rac{11,025}{32} y$ Focus: (0, 86.1) ft above vertex

Table: Example Values for Parabola $x = -2y^2 + 4y - 3$

x	y
-3	0
-3	2
-1	1

Additional info: These notes cover the foundational concepts and equations for circles and parabolas, including graphing techniques, completing the square, and real-world applications. The examples and table provide practical context for understanding the algebraic and geometric properties of conic sections.