Comprehensive Guide to Conic Sections: Circles, Parabolas, Ellipses, and Hyperbolas

Conceptual Definition: A circle is geometrically defined as the set of all points in a plane that are at a fixed, equal distance from a specific central point. This distance is known as the radius ( $r$ ).
The Standard Form Equation: The mathematical representation of a circle on a coordinate plane is given by the formula: $(x - h)^2 + (y - k)^2 = r^2$
Identifying the Center: The center of the circle is located at the coordinates $(h, k)$ .
The Sign Flip Rule (Danger Zone): It is critical to remember that the signs of $(h, k)$ flip when moving from the equation to the coordinate point. For example, in the expression $(x - 3)$ , the value of $h$ is actually $+3$ . Conversely, in the expression $(y + 2)$ , the value of $k$ is $-2$ .
Calculating the Radius: The value on the right-hand side of the equals sign is not the radius itself, but rather the square of the radius ( $r^2$ ).
The Calculation Rule (Danger Zone): To find the actual radius of the circle, you must always take the square root of the number provided on the right side of the equation.
Applied Example (Problem 4):
- Given Equation: $(x - 3)^2 + (y + 2)^2 = 4$
- Center: Following the sign flip rule, the center is $(3, -2)$ .
- Radius: The radius is calculated as $ext{radius} = ext{sqrt}(4) = 2$ .

Conceptual Definition: A parabola is a symmetrical open curve. Parabolas in conic sections can open up, down, left, or right.
Vertex: The turning point of the curve where it changes direction.
Focus: A specific point located inside the "bowl" of the parabola.
Directrix: A fixed line located behind the "bowl" of the parabola.
Determining Orientation:
- Vertical Parabola: If the $x$ variable is squared ( $y = ax^2$ ), it opens vertically—either up or down.
- Horizontal Parabola: If the $y$ variable is squared ( $x = ay^2$ ), it opens horizontally—either left or right.
The Parametric Constant (p):
- The standard math template for a parabola uses the fraction $rac{1}{4p}$ as the coefficient in front of the squared variable.
- The value of $p$ represents the exact distance from the Vertex to the Focus and simultaneously the distance from the Vertex to the Directrix.
Applied Example (Problem 6):
- Given Equation: $y = - rac{1}{20}x^2$
- Direction: The variable $x$ is squared, and the coefficient is negative, meaning the parabola opens down.
- Solving for p: Set the denominator of the coefficient equal to $4p$ : $20 = 4p$ leading to $p = 5$ .
- Key Coordinates and Lines:
  - Vertex: $(0, 0)$
  - Focus: Starting at the vertex and moving down $5$ units results in the focus at $(0, -5)$ .
  - Directrix: Starting at the vertex and moving up $5$ units results in the horizontal line equation $y = 5$ .

Conceptual Definition: An ellipse is an oval shape. Unlike a circle, which has a single radius, an ellipse has two main axes.
Major Axis: The long direction of the ellipse.
Minor Axis: The short direction of the ellipse.
The Standard Form Equation: $rac{x^2}{a^2} + rac{y^2}{b^2} = 1$
Identifying the Orientation (Rule #1):
- The variable $a^2$ is always the larger number in the denominators.
- Horizontal Ellipse: If the larger number ( $a^2$ ) is underneath the $x^2$ term, it is a wide oval.
- Vertical Ellipse: If the larger number ( $a^2$ ) is underneath the $y^2$ term, it is a tall oval.
Locating the Foci: Foci are the inner focal points of the ellipse.
Foci Formula: $c^2 = a^2 - b^2$
Note on Sign: This formula uses a minus sign because an ellipse is a closed figure, similar to the Pythagorean theorem's modified form.
Applied Example (Problem 9):
- Given Equation: $rac{x^2}{81} + rac{y^2}{225} = 1$
- Comparison: Since $225$ is larger than $81$ , then $a^2 = 225$ .
- Orientation: Because the larger number ( $225$ ) is under the $y^2$ term, the ellipse is a tall, vertical oval.

Conceptual Definition: Hyperbolas consist of two separate curves that look like parabolas facing away from each other (back-to-back bows).
Standard Form Equations:
- Opening Left/Right: $rac{x^2}{a^2} - rac{y^2}{b^2} = 1$
- Opening Up/Down: $rac{y^2}{a^2} - rac{x^2}{b^2} = 1$
Identifying the Leading Term (Rule #1):
- In a hyperbola, $a^2$ is not necessarily the largest number. Instead, $a^2$ is defined as the denominator of the first fraction (the positive fraction).
Determining Orientation:
- Horizontal Hyperbola: If the $x^2$ term is the lead fraction, the hyperbola opens left and right.
- Vertical Hyperbola: If the $y^2$ term is the lead fraction, the hyperbola opens up and down.
Locating the Foci:
Foci Formula: $c^2 = a^2 + b^2$
Note on Sign: This formula uses a plus sign because the foci of a hyperbola are located further out than the vertices.