Conic Sections and Nonlinear Systems: Study Notes

Conic Sections and Nonlinear Systems

Introduction

This chapter explores conic sections—parabolas, circles, ellipses, and hyperbolas—by analyzing their general equations, characteristics, and graphical representations. It also covers the concept of eccentricity and methods for solving nonlinear systems of equations and inequalities.

Conic Sections

General Form of Conic Sections

Conic sections are described by the general second-degree equation:

$Ax^2 + Dx + Cy^2 + Ey + F = 0$

where either $A$ or $C$ must be nonzero. The type of conic is determined by the coefficients $A$ and $C$.

Conic Section	Characteristic	Example
Parabola	Either $A = 0$ or $C = 0$, but not both	$y = x^2$
Circle	$A = C \neq 0$	$x^2 + y^2 = 16$
Ellipse	$A \neq C$, $AC > 0$	$\frac{x^2}{16} + \frac{y^2}{9} = 1$
Hyperbola	$AC < 0$	$x^2 - y^2 = 1$

Identifying Conic Sections from Equations

• Parabola: Only one variable is squared. Example: $x^2 - 6x + 8y - 7 = 0$

• Circle: $A = C \neq 0$. Example: $x^2 - 8x + y^2 + 10y = -41$

• Ellipse: $A \neq C$, $AC > 0$. Example: $4x^2 - 16x + 9y^2 + 54y = -61$

• Hyperbola: $AC < 0$. Example: $x^2 = 25 + 5y^2$

Completing the Square

To rewrite conic equations in standard form, complete the square for $x$ and $y$ terms as needed. This helps identify the center, vertices, and orientation of the conic.

Examples

• Hyperbola: $x^2 = 25 + 5y^2$ becomes $\frac{x^2}{25} - \frac{y^2}{5} = 1$

• Circle: $x^2 - 8x + y^2 + 10y = -41$ becomes $(x-4)^2 + (y+5)^2 = 0$ (a point at (4, -5))

• Ellipse: $4x^2 - 16x + 9y^2 + 54y = -61$ becomes $\frac{(x-2)^2}{9} + \frac{(y+3)^2}{4} = 1$

• Parabola: $x^2 - 6x + 8y - 7 = 0$ becomes $(x-3)^2 = -8(y-2)$ (vertex at (3, 2), opens downward)

• Hyperbola: $4y^2 - 16y - 9x^2 + 18x = -43$ becomes $\frac{(y-2)^2}{4} - \frac{(x-1)^2}{9} = 1$ (center at (1, 2))

Conic Definitions and Eccentricity

Definition of a Conic

A conic is the set of all points $P(x, y)$ in a plane such that the ratio of the distance from $P$ to a fixed point (focus) and the distance from $P$ to a fixed line (directrix) is constant. This constant ratio is called the eccentricity, $e$.

Eccentricity of Ellipses and Hyperbolas

• For ellipses: $a^2 > b^2$ and $c = \sqrt{a^2 - b^2}$

• $0 < c < a$

• $0 < e = \frac{c}{a} < 1$

• For hyperbolas: $e > 1$

• Ellipses with $e$ close to 0 are nearly circular.

Examples: Finding Eccentricity

• Ellipse: $\frac{x^2}{9} + \frac{y^2}{16} = 1$ $a^2 = 16$, $b^2 = 9$, $a = 4$, $c = \sqrt{16-9} = \sqrt{7}$ $e = \frac{\sqrt{7}}{4} \approx 0.66$

• Ellipse: $5x^2 + 10y^2 = 50$ $\frac{x^2}{10} + \frac{y^2}{5} = 1$, $a^2 = 10$, $b^2 = 5$, $a = \sqrt{10}$, $c = \sqrt{10-5} = \sqrt{5}$ $e = \frac{\sqrt{5}}{\sqrt{10}} = \frac{1}{\sqrt{2}} \approx 0.71$

• Hyperbola: $\frac{x^2}{9} - \frac{y^2}{4} = 1$ $a^2 = 9$, $b^2 = 4$, $a = 3$, $c = \sqrt{9+4} = \sqrt{13}$ $e = \frac{\sqrt{13}}{3} \approx 1.2$

Comparing Eccentricities

As $e$ increases, ellipses become more elongated and hyperbolas become more open. For circles, $e = 0$.

Finding Equations of Conics Using Eccentricity

• Given focus at (3, 0) and $e = 2$: Hyperbola, $c = 3$, $e = \frac{c}{a} \Rightarrow a = \frac{3}{2}$, $b^2 = c^2 - a^2 = 9 - (\frac{9}{4}) = \frac{27}{4}$ Equation: $\frac{x^2}{(3/2)^2} - \frac{y^2}{(\sqrt{27}/2)^2} = 1$ or $\frac{x^2}{9/4} - \frac{y^2}{27/4} = 1$

• Given vertex at (0, -8) and $e = \frac{1}{2}$: Ellipse, $a = 8$, $e = \frac{c}{a} \Rightarrow c = 4$, $b^2 = a^2 - c^2 = 64 - 16 = 48$ Equation: $\frac{y^2}{64} + \frac{x^2}{48} = 1$

Application: Planetary Orbits

The orbit of Mars is an ellipse with the sun at one focus. Given $e = 0.0935$ and closest distance $a - c = 128.5$ million miles:

• $e = \frac{c}{a}$, $c = a - 128.5$

• $0.0935a = a - 128.5$

• $a \approx 141.8$, $c \approx 13.3$

• Maximum distance: $a + c \approx 155.1$ million miles

Nonlinear Systems

Solving Nonlinear Systems by Elimination

To solve systems involving conic sections, use elimination or substitution to reduce the system to a single variable.

• Example: $x^2 + y^2 = 4$, $2x^2 - y^2 = 8$ Add: $3x^2 = 12 \Rightarrow x = 2$ or $x = -2$ Substitute into first equation: $y = 0$ Solution set: $(2, 0), (-2, 0)$

Solving by Combination of Methods

• Example: $x^2 + 3xy + y^2 = 22$, $x^2 - xy + y^2 = 6$ Subtract: $4xy = 16 \Rightarrow y = \frac{4}{x}$ Substitute into one equation and solve for $x$ and $y$.

• Solution set: $(\sqrt{2}, 2\sqrt{2}), (-\sqrt{2}, -2\sqrt{2}), (2\sqrt{2}, \sqrt{2}), (-2\sqrt{2}, -\sqrt{2})$

Graphing Nonlinear Systems of Inequalities

• Example: $\frac{x^2}{25} + \frac{y^2}{9} \leq 1$, $\frac{x^2}{4} - \frac{y^2}{9} \geq 1$

• The solution is the region inside the ellipse and outside the hyperbola, as shown by the overlap of shaded regions on the graph.

Summary Table: Conic Section Characteristics

Conic	Standard Form	Key Features
Parabola	$(x-h)^2 = 4p(y-k)$ or $(y-k)^2 = 4p(x-h)$	Vertex, focus, directrix
Circle	$(x-h)^2 + (y-k)^2 = r^2$	Center, radius
Ellipse	$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$	Center, vertices, foci, axes
Hyperbola	$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$	Center, vertices, foci, asymptotes