Analytic Geometry: Hyperbolas and Parabolas

Analytic Geometry: Conic Sections

Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types are parabolas, hyperbolas, ellipses, and circles. In College Algebra, we focus on their equations, properties, and graphs.

• Parabola: Defined by a quadratic equation with one squared term.

• Hyperbola: Defined by a quadratic equation with two squared terms of opposite signs.

Hyperbolas

Standard Form and Identification

A hyperbola is the set of all points where the difference of the distances to two fixed points (foci) is constant. The general equation is:

• Standard Form (centered at (h, k)):

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$   (opens left/right) $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$   (opens up/down)

• Key Features:

  • Center: $(h, k)$

  • Vertices: $a$ units from the center along the transverse axis

  • Co-vertices: $b$ units from the center along the conjugate axis

  • Asymptotes: Lines the hyperbola approaches but never touches

Example 1: Finding the Standard Form

Given: $4x^2 - 9y^2 - 8x + 36y - 68 = 0$

1. Group and rearrange terms: $4x^2 - 8x - 9y^2 + 36y = 68$

2. Complete the square for $x$ and $y$:

   • $4(x^2 - 2x + 1) - 9(y^2 - 4y + 4) = 68 + 4 - 36$

   • $4(x-1)^2 - 9(y-2)^2 = 36$

3. Divide by 36:

   • $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{4} = 1$

4. Center: $(1,2)$, a = 3, b = 2

Example 2: Hyperbola Opening Up/Down

Given: $9y^2 - 25x^2 + 72y - 100x - 181 = 0$

1. Group and rearrange terms: $9y^2 + 72y - 25x^2 - 100x = 181$

2. Complete the square:

   • $9(y^2 + 8y + 16) - 25(x^2 + 4x + 4) = 181 + 144 - 100$

   • $9(y+4)^2 - 25(x+2)^2 = 225$

3. Divide by 225:

   • $\frac{(y+4)^2}{25} - \frac{(x+2)^2}{9} = 1$

4. Center: $(−2,−4)$, a = 5, b = 3

Graphing Hyperbolas

• Draw the center at $(h, k)$

• Mark vertices $a$ units from the center along the transverse axis

• Mark co-vertices $b$ units from the center along the conjugate axis

• Draw asymptotes through the center with slopes $\pm \frac{a}{b}$ or $\pm \frac{b}{a}$ depending on orientation

• Sketch the two branches approaching the asymptotes

Asymptotes Equations:

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

  • $y = k \pm \frac{b}{a}(x-h)$

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

  • $y = k \pm \frac{a}{b}(x-h)$

Visual Representation

Hyperbolas have two branches, each approaching their respective asymptotes. The orientation (horizontal or vertical) depends on which variable is positive in the standard form.

Parabolas

Standard Form and Identification

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The general equation is:

• Standard Form (vertical axis): $y = a(x-h)^2 + k$

• Standard Form (horizontal axis): $x = a(y-k)^2 + h$

• Key Features:

  • Vertex: $(h, k)$

  • Axis of symmetry: $x = h$ (vertical), $y = k$ (horizontal)

  • Direction: Opens up/down or left/right depending on the sign of $a$

Example 1: Vertical Parabola

Given: $x^2 + 4x - 5y - 11 = 0$

1. Group and rearrange terms: $x^2 + 4x = 5y + 11$

2. Complete the square:

   • $(x^2 + 4x + 4) = 5y + 11 + 4$

   • $(x+2)^2 = 5y + 15$

   • $(x+2)^2 = 5(y + 3)$

   • $y = \frac{1}{5}(x+2)^2 - 3$

3. Vertex: $(−2,−3)$

4. Opens upward (since coefficient of $y$ is positive)

Example 2: Horizontal Parabola

Given: $y^2 + 2y + 2x + 7 = 0$

1. Group and rearrange terms: $y^2 + 2y = -2x - 7$

2. Complete the square:

   • $(y^2 + 2y + 1) = -2x - 7 + 1$

   • $(y+1)^2 = -2x - 6$

   • $(y+1)^2 = -2(x + 3)$

   • $x = -\frac{1}{2}(y+1)^2 - 3$

3. Vertex: $(−3,−1)$

4. Opens left (since coefficient of $x$ is negative)

Example 3: Another Vertical Parabola

Given: $x^2 - 6x + 2y + 13 = 0$

1. Group and rearrange terms: $x^2 - 6x = -2y - 13$

2. Complete the square:

   • $(x^2 - 6x + 9) = -2y - 13 + 9$

   • $(x-3)^2 = -2y - 4$

   • $(x-3)^2 = -2(y + 2)$

   • $y = -\frac{1}{2}(x-3)^2 - 2$

3. Vertex: $(3,−2)$

4. Opens downward (since coefficient of $y$ is negative)

Graphing Parabolas

• Plot the vertex $(h, k)$

• Draw the axis of symmetry

• Sketch the curve opening in the direction indicated by the sign of $a$

Summary Table: Conic Section Equations

Conic Section	Standard Equation	Key Features
Parabola (vertical)	$y = a(x-h)^2 + k$	Vertex $(h, k)$, opens up/down
Parabola (horizontal)	$x = a(y-k)^2 + h$	Vertex $(h, k)$, opens left/right
Hyperbola (horizontal)	$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$	Center $(h, k)$, opens left/right, asymptotes $y = k \pm \frac{b}{a}(x-h)$
Hyperbola (vertical)	$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$	Center $(h, k)$, opens up/down, asymptotes $y = k \pm \frac{a}{b}(x-h)$

Key Takeaways

• Completing the square is essential for converting general conic equations to standard form.

• Identifying the center/vertex and orientation helps in graphing conic sections accurately.

• Asymptotes guide the shape of hyperbolas, while the axis of symmetry guides parabolas.

Additional info: The notes include step-by-step algebraic manipulation, graph sketches, and identification of conic section features, suitable for College Algebra students studying analytic geometry.