📐Algebra and Trigonometry Unit 12 – Analytic Geometry

What's This Unit All About?

• Analytic geometry combines algebra and geometry to solve problems involving geometric shapes on a coordinate plane
• Focuses on representing geometric objects using equations and coordinates
• Enables the study of geometric properties using algebraic methods
• Provides a foundation for calculus and other advanced mathematics
• Applies to various fields such as physics, engineering, and computer graphics
• Helps develop problem-solving skills and mathematical reasoning
• Encourages the use of multiple representations (graphical, algebraic, and numerical) to analyze geometric relationships

Key Concepts and Definitions

• Coordinate plane: a two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis
• Origin: the point where the x-axis and y-axis intersect, represented by the coordinates (0, 0)
• Ordered pair: a pair of numbers (x, y) that represents the position of a point on the coordinate plane
  • x-coordinate: the first number in an ordered pair, representing the horizontal distance from the origin
  • y-coordinate: the second number in an ordered pair, representing the vertical distance from the origin
• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, used to find the distance between two points on the coordinate plane
• Midpoint formula: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$, used to find the coordinates of the midpoint between two points
• Slope: the steepness of a line, calculated as the change in y-coordinates divided by the change in x-coordinates, $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Parallel lines: lines that have the same slope and never intersect
• Perpendicular lines: lines that intersect at a 90-degree angle and have slopes that are negative reciprocals of each other

Coordinate Systems: The Playing Field

• Cartesian coordinate system: the most common coordinate system, using perpendicular x and y axes to define points in a plane
• Polar coordinate system: represents points using an angle and a distance from a fixed point called the pole
  • Polar coordinates: $(r, \theta)$, where $r$ is the distance from the pole and $\theta$ is the angle from the polar axis
• Conversion between Cartesian and polar coordinates:
  • $x = r \cos(\theta)$
  • $y = r \sin(\theta)$
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$
• Three-dimensional coordinate system: adds a third perpendicular axis (z-axis) to represent points in 3D space
• Spherical and cylindrical coordinate systems: alternative 3D coordinate systems used in specific applications
• Choosing the appropriate coordinate system depends on the problem and the symmetry of the geometric objects involved

Lines and Their Equations

• Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• Point-slope form: $y - y_1 = m(x - x_1)$, used when given a point on the line and the slope
• Standard form: $Ax + By = C$, where $A$, $B$, and $C$ are constants and $A$ and $B$ are not both zero
• Vertical lines: have an undefined slope and are represented by the equation $x = a$, where $a$ is a constant
• Horizontal lines: have a slope of zero and are represented by the equation $y = b$, where $b$ is a constant
• Parallel and perpendicular lines:
  • Parallel lines have the same slope
  • Perpendicular lines have slopes that are negative reciprocals of each other
• Distance from a point to a line: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$, where $(x_0, y_0)$ is the point and $Ax + By + C = 0$ is the line equation in general form

Circles: Round and Round We Go

• Standard form of a circle equation: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius
• General form of a circle equation: $x^2 + y^2 + Dx + Ey + F = 0$, where $D$, $E$, and $F$ are constants
• Converting between standard and general form:
  • From general to standard: complete the square for both $x$ and $y$ terms
  • From standard to general: expand the squared binomials and combine like terms
• Finding the center and radius from the general form equation:
  • Center: $(-\frac{D}{2}, -\frac{E}{2})$
  • Radius: $\sqrt{(\frac{D}{2})^2 + (\frac{E}{2})^2 - F}$
• Tangent lines to circles: perpendicular to the radius at the point of tangency
• Inscribed angles: an angle formed by two chords with a common endpoint on the circle; it is half the measure of the central angle that subtends the same arc
• Secant lines: lines that intersect a circle at two points
• Chord: a line segment with endpoints on the circle

Conic Sections: Slicing Things Up

• Conic sections: curves formed by the intersection of a plane with a double cone
• Ellipse: the set of points in a plane such that the sum of the distances from two fixed points (foci) is constant
  • Standard form of an ellipse equation: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ is the center, $a$ and $b$ are the semi-major and semi-minor axes
• Hyperbola: the set of points in a plane such that the difference of the distances from two fixed points (foci) is constant
  • Standard form of a hyperbola equation: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (horizontal) or $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ (vertical)
• Parabola: the set of points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix)
  • Standard form of a parabola equation: $y = a(x - h)^2 + k$ (vertical) or $x = a(y - k)^2 + h$ (horizontal), where $(h, k)$ is the vertex and $a$ determines the direction and width
• Eccentricity: a measure of how much a conic section deviates from being circular
  • Eccentricity values: circle (0), ellipse (between 0 and 1), parabola (1), hyperbola (greater than 1)

Practical Applications

• Planetary orbits: elliptical orbits can be modeled using the equations of ellipses
• Satellite dish design: parabolic shapes are used to focus signals from satellites
• Whispering galleries: elliptical rooms where a whisper at one focus can be heard clearly at the other focus
• Sundials: the shadow cast by the gnomon follows a hyperbolic path
• Suspension bridges: the shape of the main cables is a parabola
• Reflective properties of conic sections:
  • Ellipses: light rays emanating from one focus reflect off the ellipse and pass through the other focus
  • Hyperbolas: light rays directed towards one focus reflect off the hyperbola as if they originated from the other focus
  • Parabolas: light rays parallel to the axis of symmetry reflect off the parabola and pass through the focus
• Optimization problems: finding the shortest path, maximizing area, or minimizing cost using geometric constraints

Common Pitfalls and How to Avoid Them

• Confusing the equations of conic sections: memorize the standard forms and their key differences
• Forgetting to complete the square when converting circle equations: practice the process and double-check your work
• Misidentifying the center, vertices, or foci of conic sections: carefully examine the equation and sketch the graph
• Incorrectly applying the distance or midpoint formulas: pay attention to the order of the coordinates and the signs in the formulas
• Misinterpreting the slope: remember that slope is the change in y divided by the change in x, not the other way around
• Confusing parallel and perpendicular slopes: parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals
• Mixing up the equations for horizontal and vertical lines: horizontal lines have the form $y = b$, while vertical lines have the form $x = a$
• Struggling with three-dimensional coordinate systems: practice visualizing points and shapes in 3D space and break down complex problems into simpler components