Conic Sections: Circles, Ellipses, Hyperbolas, and Parabolas – Precalculus Study Notes

Conic Sections

Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four main types are circles, ellipses, hyperbolas, and parabolas. Each has a unique geometric definition and standard equation.

Circles

Finding the Equation of a Circle

A circle is the set of all points in a plane that are a fixed distance (radius) from a fixed point (center). The standard form of a circle's equation is $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Finding the Center: The center is the midpoint between two endpoints of a diameter. Use the midpoint formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Finding the Radius: The radius is the distance from the center to any point on the circle. Use the distance formula: $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Standard Form: Substitute the center and radius into the standard equation.

Example: Given endpoints of a diameter at $(2, 8)$ and $(10, 8)$:

• Center: $\left( \frac{2+10}{2}, \frac{8+8}{2} \right) = (6, 8)$

• Radius: $r = \sqrt{(10-6)^2 + (8-8)^2} = \sqrt{16} = 4$

• Equation: $(x-6)^2 + (y-8)^2 = 16$

Ellipses

Definition and Standard Equations

An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant. The standard equation for an ellipse centered at $(h, k)$ is:

• Horizontal major axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Vertical major axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

• Where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $c$ is the distance from the center to each focus, with $c^2 = a^2 - b^2$.

Key Features:

• Center: $(h, k)$

• Vertices: Endpoints of the major axis

• Foci: Located along the major axis, $c$ units from the center

• Length of major axis: $2a$

• Length of minor axis: $2b$

Example: For $\frac{(x+2)^2}{4} + \frac{(y-1)^2}{9} = 1$:

• Center: $(-2, 1)$

• Orientation: Vertical (since $b^2 > a^2$)

• Vertices: $(-2, 1 \pm 3)$

• Foci: $c = \sqrt{9-4} = \sqrt{5}$, so foci at $(-2, 1 \pm \sqrt{5})$

Hyperbolas

Definition and Standard Equations

A hyperbola is the set of all points such that the difference of the distances from two fixed points (foci) is constant. The standard equation for a hyperbola centered at $(h, k)$ is:

• Horizontal transverse axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Vertical transverse axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

• Where $a$ is the distance from the center to a vertex, $b$ is related to the slope of the asymptotes, and $c$ is the distance from the center to each focus, with $c^2 = a^2 + b^2$.

Key Features:

• Center: $(h, k)$

• Vertices: Endpoints of the transverse axis

• Foci: Located along the transverse axis, $c$ units from the center

• Asymptotes: Lines the hyperbola approaches, with equations $y-k = \pm \frac{b}{a}(x-h)$ (horizontal) or $y-k = \pm \frac{a}{b}(x-h)$ (vertical)

Example: For $\frac{(x+2)^2}{4} - \frac{(y-1)^2}{9} = 1$:

• Center: $(-2, 1)$

• Orientation: Horizontal (since $x$ term is positive)

• Vertices: $(-2 \pm 2, 1)$

• Foci: $c = \sqrt{4+9} = \sqrt{13}$, so foci at $(-2 \pm \sqrt{13}, 1)$

• Asymptotes: $y-1 = \pm \frac{3}{2}(x+2)$

Parabolas

Definition and Standard Equations

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The standard equations for parabolas with vertex at $(h, k)$ are:

• Vertical axis: $(x-h)^2 = 4p(y-k)$

• Horizontal axis: $(y-k)^2 = 4p(x-h)$

• Where $p$ is the distance from the vertex to the focus (and to the directrix).

Key Features:

• Vertex: $(h, k)$

• Focus: $(h, k+p)$ for vertical, $(h+p, k)$ for horizontal

• Directrix: $y = k-p$ for vertical, $x = h-p$ for horizontal

• Axis of symmetry: Line passing through the vertex and focus

Example: For $(x-2)^2 = 8(y+1)$:

• Vertex: $(2, -1)$

• Opens upward (since $p > 0$)

• $4p = 8 \implies p = 2$

• Focus: $(2, -1+2) = (2, 1)$

• Directrix: $y = -1-2 = -3$