Conic Sections and Partial Fraction Decomposition: Study Notes for Precalculus

Conic Sections

The Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus). Parabolas are fundamental in analytic geometry and have applications in physics and engineering.

• Definition: The locus of points P such that the distance from P to the focus equals the distance from P to the directrix.

• Standard Forms: For vertex at (0,0), focus at (a,0): $y^2 = 4ax$ or $x^2 = 4ay$

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

• Direction: The sign of a determines the direction the parabola opens (right, left, up, or down).

• Latus Rectum: The line segment through the focus, parallel to the directrix, with endpoints on the parabola. Its length is $|4a|$.

Example: Find the equation of a parabola with vertex at (0,0) and focus at (2,0): $y^2 = 8x$ Graph the equation and identify the latus rectum.

Table: Parabola Forms and Properties

Vertex	Focus	Directrix	Equation	Description
(0,0)	(a,0)	x = -a	$y^2 = 4ax$	Opens right if a > 0, left if a < 0
(0,0)	(0,a)	y = -a	$x^2 = 4ay$	Opens up if a > 0, down if a < 0

The Ellipse

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. Ellipses are important in planetary motion and optics.

• Definition: The locus of points P such that $PF_1 + PF_2 = d$, where $F_1$ and $F_2$ are the foci.

• Standard Form (center at origin): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (major axis along x-axis) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (major axis along y-axis)

• Vertices: $(\pm a, 0)$ or $(0, \pm a)$

• Foci: $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 - b^2$

Example: Find the equation of an ellipse with center at (0,0), vertices at (\pm 5, 0), and foci at (\pm 3, 0): $\frac{x^2}{25} + \frac{y^2}{16} = 1$

Table: Ellipse Forms and Properties

Center	Major Axis	Vertices	Foci	Equation
(h,k)	Parallel to x-axis	(h\pm a, k)	(h\pm c, k)	$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
(h,k)	Parallel to y-axis	(h, k\pm a)	(h, k\pm c)	$\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

The Hyperbola

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (the foci) is constant. Hyperbolas appear in navigation and physics.

• Definition: The locus of points P such that $|PF_1 - PF_2| = d$.

• Standard Form (center at origin): $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (transverse axis along x-axis) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (transverse axis along y-axis)

• Vertices: $(\pm a, 0)$ or $(0, \pm a)$

• Foci: $(\pm c, 0)$ or $(0, \pm c)$, where $c^2 = a^2 + b^2$

• Asymptotes: Lines the hyperbola approaches as $x$ or $y$ becomes large. For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, asymptotes are $y = \pm \frac{b}{a}x$.

Example: Analyze the equation $4x^2 - 9y^2 = 36$ and find its asymptotes. Rewrite as $\frac{x^2}{9} - \frac{y^2}{4} = 1$, so asymptotes are $y = \pm \frac{2}{3}x$.

Table: Hyperbola Forms and Properties

Center	Transverse Axis	Vertices	Foci	Equation	Asymptotes
(h,k)	Parallel to x-axis	(h\pm a, k)	(h\pm c, k)	$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$	$y = k \pm \frac{b}{a}(x-h)$
(h,k)	Parallel to y-axis	(h, k\pm a)	(h, k\pm c)	$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$	$y = k \pm \frac{a}{b}(x-h)$

Partial Fraction Decomposition

Introduction to Partial Fractions

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This is useful for integration and solving equations in calculus and algebra.

• Case 1: Q has only nonrepeated linear factors If $Q(x)$ has distinct linear factors, decompose $\frac{P(x)}{Q(x)}$ as $\frac{A_1}{x - r_1} + \frac{A_2}{x - r_2} + \cdots$

• Case 2: Q has repeated linear factors If $Q(x)$ has a repeated factor $(x - r)^n$, include terms $\frac{A_1}{x - r} + \frac{A_2}{(x - r)^2} + \cdots + \frac{A_n}{(x - r)^n}$

• Case 3: Q contains a nonrepeated irreducible quadratic factor Include terms of the form $\frac{Ax + B}{x^2 + bx + c}$

• Case 4: Q contains a repeated irreducible quadratic factor Include terms $\frac{A_1x + B_1}{x^2 + bx + c} + \frac{A_2x + B_2}{(x^2 + bx + c)^2} + \cdots$

Example: Find the partial fraction decomposition of $\frac{4}{x^2 - x}$. $\frac{4}{x^2 - x} = \frac{A}{x} + \frac{B}{x - 1}$

Table: Partial Fraction Decomposition Cases

Case	Form of Q(x)	Decomposition Terms
1	Distinct linear factors	$\frac{A_1}{x - r_1} + \frac{A_2}{x - r_2} + \cdots$
2	Repeated linear factors	$\frac{A_1}{x - r} + \frac{A_2}{(x - r)^2} + \cdots$
3	Irreducible quadratic factor	$\frac{Ax + B}{x^2 + bx + c}$
4	Repeated irreducible quadratic factor	$\frac{A_1x + B_1}{x^2 + bx + c} + \frac{A_2x + B_2}{(x^2 + bx + c)^2} + \cdots$

Summary

• Conic sections include parabolas, ellipses, and hyperbolas, each defined by a unique geometric property and standard equation.

• Partial fraction decomposition is a method for breaking down rational expressions into simpler terms, facilitating further algebraic manipulation and integration.

Additional info: These notes expand on the definitions, equations, and properties of conic sections and partial fraction decomposition, providing context and examples for Precalculus students.