Conic Sections: Parabolas and Ellipses – Study Notes for Precalculus

Conic Sections

Introduction

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The most common conic sections studied in Precalculus are parabolas, ellipses, hyperbolas, and circles. This guide focuses on parabolas and ellipses, their equations, properties, and how to find key features such as vertices, foci, and axes of symmetry.

Parabolas

Standard Forms and Properties

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The orientation and position of a parabola depend on its equation.

• Standard Form (Horizontal Axis): $(y - k)^2 = 4a(x - h)$

• Standard Form (Vertical Axis): $(x - h)^2 = 4a(y - k)$

• Vertex: $(h, k)$

• Axis of Symmetry: The line passing through the vertex and focus

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

• Directrix: $x = h - a$ for horizontal; $y = k - a$ for vertical

• Latus Rectum: A line segment through the focus perpendicular to the axis of symmetry; its length is $|4a|$

Examples and Applications

• Example 1: Find the equation of a parabola with vertex $(0,0)$, focus $(3,0)$, opening to the right.

  • Since it opens right, use $(y - k)^2 = 4a(x - h)$

  • $a = 3$

  • Equation: $y^2 = 12x$

  • Latus Rectum endpoints: $(3, 2)$ and $(3, -2)$

• Example 2: Vertex $(1,3)$, focus $(5,3)$, opens right.

  • $a = 4$

  • Equation: $(y - 3)^2 = 16(x - 1)$

  • Latus Rectum endpoints: $(5, 5)$ and $(5, 1)$

• Example 3: Vertex $(4,2)$, opens upward.

  • Use $(x - h)^2 = 4a(y - k)$

  • Plug in focus and vertex to solve for $a$

  • Equation: $(x - 4)^2 = 4a(y - 2)$

  • Latus Rectum endpoints: $(4, 2 + a)$ and $(4, 2 - a)$

Graphing Parabolas

• Identify the vertex and focus from the graph.

• Determine the direction the parabola opens (right, left, up, down).

• Use the standard form to write the equation.

• Example: Vertex $(-7,0)$, focus $(10,14)$, opens to the right.

  • Equation: $(y - k)^2 = 4a(x - h)$

  • Find $a$ using the distance between vertex and focus.

  • Equation: $y^2 = 28(x + 7)$

• Example: Vertex $(-1, -3)$, focus $(0, -4)$, opens to the left.

  • Equation: $(y - k)^2 = -4a(x - h)$

  • Equation: $(y + 3)^2 = -4(x + 1)$

Ellipses

Standard Forms and Properties

An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant. The major axis is the longest diameter, and the minor axis is the shortest.

• Standard Form (Horizontal Major Axis): $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Standard Form (Vertical Major Axis): $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

• Center: $(h, k)$

• Vertices: $(h \pm a, k)$ for horizontal; $(h, k \pm a)$ for vertical

• Foci: $(h \pm c, k)$ for horizontal; $(h, k \pm c)$ for vertical, where $c = \sqrt{a^2 - b^2}$

• Major Axis Length: $2a$

• Minor Axis Length: $2b$

Examples and Applications

• Example 1: $\frac{x^2}{16} + \frac{y^2}{4} = 1$ (center at $(0,0)$, horizontal major axis)

  • $a^2 = 16$, $b^2 = 4$

  • $a = 4$, $b = 2$

  • $c = \sqrt{16 - 4} = \sqrt{12} \approx 3.46$

  • Vertices: $(4,0)$ and $(-4,0)$

  • Foci: $(3.46,0)$ and $(-3.46,0)$

• Example 2: $\frac{y^2}{16} + \frac{x^2}{4} = 1$ (center at $(0,0)$, vertical major axis)

  • $a^2 = 16$, $b^2 = 4$

  • $a = 4$, $b = 2$

  • $c = \sqrt{16 - 4} = \sqrt{12} \approx 3.46$

  • Vertices: $(0,4)$ and $(0,-4)$

  • Foci: $(0,3.46)$ and $(0,-3.46)$

• Example 3: $\frac{(x - 2)^2}{4} + \frac{(y - 0)^2}{36} = 1$ (center at $(2,0)$, vertical major axis)

  • $a^2 = 36$, $b^2 = 4$

  • $a = 6$, $b = 2$

  • $c = \sqrt{36 - 4} = \sqrt{32} \approx 5.66$

  • Vertices: $(2,6)$ and $(2,-6)$

  • Foci: $(2,5.66)$ and $(2,-5.66)$

Summary Table: Ellipse Properties

Equation	Center	Vertices	Foci	Major Axis	Minor Axis
$\frac{x^2}{16} + \frac{y^2}{4} = 1$	$(0,0)$	$(4,0), (-4,0)$	$(3.46,0), (-3.46,0)$	Horizontal	Vertical
$\frac{y^2}{16} + \frac{x^2}{4} = 1$	$(0,0)$	$(0,4), (0,-4)$	$(0,3.46), (0,-3.46)$	Vertical	Horizontal
$\frac{(x-2)^2}{4} + \frac{(y-0)^2}{36} = 1$	$(2,0)$	$(2,6), (2,-6)$	$(2,5.66), (2,-5.66)$	Vertical	Horizontal

Key Terms

• Vertex: The point where the parabola or ellipse is centered or turns.

• Focus (Foci): Special points used to define conic sections.

• Directrix: A fixed line used in the definition of a parabola.

• Latus Rectum: A line segment perpendicular to the axis of symmetry of a parabola, passing through the focus.

• Major Axis: The longest diameter of an ellipse.

• Minor Axis: The shortest diameter of an ellipse.

Additional info:

• These notes are based on handwritten class notes and diagrams, with some context inferred for completeness.

• For hyperbolas and circles, similar principles apply but are not covered in this guide.