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12.4 Rotation of Axes

2 min read•june 24, 2024

are fascinating curves formed by slicing a cone. They include circles, ellipses, parabolas, and hyperbolas. Each has unique properties and can be described by specific equations, which we'll learn to identify and manipulate.

Understanding conic sections is crucial for many real-world applications. From satellite orbits to architectural design, these curves play a vital role. We'll explore how to analyze and transform them, unlocking their potential in various fields.

Conic Sections

General form of conic sections

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  • of a conic section Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
    • Coefficients AA, BB, and CC cannot all be zero at the same time
  • Conic sections classified based on the B2−4ACB^2 - 4AC
    • when B2−4AC<0B^2 - 4AC < 0 (circle, oval)
    • when B2−4AC=0B^2 - 4AC = 0 (U-shaped curve)
    • when B2−4AC>0B^2 - 4AC > 0 (two separate curved parts)
  • Circle is a special case of an ellipse where B=0B = 0 and A=CA = C

Rotation of axes for conics

  • eliminates the xyxy term in the general form equation
  • θ\theta calculated using tan⁡2θ=BA−C\tan 2\theta = \frac{B}{A-C}
    • Choose angle between 0∘0^\circ and 90∘90^\circ
  • for rotation of axes:
    • x=x′cos⁡θ−y′sin⁡θx = x'\cos\theta - y'\sin\theta
    • y=x′sin⁡θ+y′cos⁡θy = x'\sin\theta + y'\cos\theta
  • Substitute formulas, expand, and simplify to obtain
  • is essential for calculating the rotation angle and performing the axis transformation

Standard form of rotated conics

  • Rotated form of a conic section A′x′2+C′y′2+D′x′+E′y′+F′=0A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0
  • for both x′x' and y′y' terms:
    1. Divide equation by constant term of x′2x'^2 (or y′2y'^2 if x′2x'^2 term missing)
    2. Move constant terms to right side of equation
    3. Factor out coefficients of x′x' and y′y'
    4. Add square of half the coefficient of x′x' (or y′y') to both sides
  • Rewrite equation in by renaming variables and simplifying

Analysis of non-rotated conics

  • Identify conic section type using B2−4ACB^2 - 4AC
  • For ellipses and hyperbolas:
    • (h,k)(h, k) found by completing square for xx and yy terms
    • Lengths of major and minor axes determined using coefficients of x2x^2 and y2y^2
    • calculated to measure deviation from a circle (0 for circle, between 0 and 1 for ellipse, greater than 1 for hyperbola)
  • For parabolas:
    • identified by completing square for variable with squared term
    • Direction of opening based on sign of coefficient of squared term (positive opens upward, negative opens downward)
    • and found using equation

Mathematical Foundations

  • : The 2D system where conic sections are graphed and analyzed
  • : Form the basis for describing conic sections mathematically
  • : Provides tools for transforming conic sections, including rotation and translation of axes

Key Terms to Review (39)

Angle of Rotation: The angle of rotation refers to the amount of angular displacement or the change in the orientation of an object or coordinate system around a fixed axis. It is a fundamental concept in the study of rotational motion and coordinate transformations.
Center: The center of a geometric shape is the point that is equidistant from all points on the shape's perimeter or boundary. It is a crucial concept in understanding the properties and equations of various conic sections, including the ellipse, hyperbola, and the effects of rotating the axes of these shapes.
Center of a hyperbola: The center of a hyperbola is the midpoint of the line segment joining its two foci. It is also the point where the transverse and conjugate axes intersect.
Center of an ellipse: The center of an ellipse is the midpoint of both the major and minor axes, serving as the point of symmetry for the ellipse. It is typically denoted by a coordinate pair $(h, k)$ in the Cartesian plane.
Co-vertex: The co-vertices of an ellipse are the endpoints of the minor axis. They are perpendicular to and lie at the midpoint of the major axis.
Complete the Square: Completing the square is a mathematical technique used to transform a quadratic expression in the form $ax^2 + bx + c$ into the standard form $a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. This method is particularly useful when solving quadratic equations and graphing parabolas.
Conic Sections: Conic sections are the curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola, and they have important applications in various fields, including mathematics, physics, and engineering.
Coordinate plane: A coordinate plane is a two-dimensional surface formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These axes divide the plane into four quadrants used for graphing equations and geometric shapes.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional plane that uses a horizontal (x-axis) and a vertical (y-axis) line to define the position of a point. It provides a way to represent and analyze relationships between variables in a visual and mathematical manner.
Degenerate conic sections: Degenerate conic sections are special cases of conic sections that do not form the usual shapes like ellipses, parabolas, or hyperbolas. They occur when the plane intersects the cone at its vertex or in other ways that produce a single point, a line, or intersecting lines.
Directrix: A directrix is a fixed line used in the geometric definition of a conic section. For a parabola, it is equidistant from any point on the curve to the focus and the directrix.
Directrix: The directrix is a fixed line used in the definition of a conic section, such as an ellipse, hyperbola, or parabola. It serves as a reference point in the geometric construction and mathematical description of these curves.
Discriminant: The discriminant is a value calculated from the coefficients of a quadratic equation $ax^2 + bx + c = 0$. It determines the nature and number of roots of the quadratic equation.
Discriminant: The discriminant is a value that determines the nature of the solutions to a quadratic equation. It provides information about the number and type of solutions, and is a crucial concept in the study of quadratic functions and the rotation of axes.
Eccentricity: Eccentricity measures the deviation of a conic section from being circular. It is denoted by $e$ and determines the shape of the conic.
Eccentricity: Eccentricity is a measure of the shape or deviation of a conic section from a perfect circle. It is a dimensionless quantity that describes the elongation or flattening of a conic section, such as an ellipse, hyperbola, or parabola, and is a fundamental property that characterizes these geometric shapes.
Ellipse: An ellipse is a closed, two-dimensional shape that resembles an elongated circle. It is one of the fundamental conic sections, which are the shapes formed by the intersection of a plane and a cone.
Focus: A focus (plural: foci) is a point used to define and describe conic sections such as ellipses, parabolas, and hyperbolas. In these shapes, distances to the focus have special geometric properties.
Focus: The focal point or point of concentration, the center of interest or activity. In the context of conic sections, the focus refers to a specific point that defines the shape and properties of these geometric figures.
General Form: The general form of an equation is a standardized way of expressing the equation that reveals its underlying structure and characteristics. This term is particularly relevant in the context of various mathematical functions and conic sections, as it allows for a concise and informative representation of these entities.
Hyperbola: A hyperbola is a type of conic section, which is a curve formed by the intersection of a plane and a cone. It is characterized by two symmetric branches that open in opposite directions and are connected by a center point.
Linear Algebra: Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, matrices, and the properties of linear transformations. It provides a framework for analyzing and solving systems of linear equations, which are fundamental in many areas of science, engineering, and mathematics.
Major axis: The major axis of an ellipse is the longest diameter, passing through its center and both foci. It defines the maximum extent of the ellipse.
Major Axis: The major axis of an ellipse is the longest diameter or the line segment that passes through the center of the ellipse and connects the two points on the ellipse that are farthest apart. It is a fundamental property of the ellipse that is crucial in understanding the geometry and behavior of this conic section.
Minor axis: The minor axis of an ellipse is the shortest diameter that passes through the center and is perpendicular to the major axis. It bisects the ellipse into two equal halves along its shortest dimension.
Minor Axis: The minor axis of an ellipse is the shorter of the two perpendicular axes that define the shape of the ellipse. It represents the shortest distance across the ellipse and is perpendicular to the major axis, which is the longer of the two axes.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Quadratic Equations: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratic equations are central to the study of systems of nonlinear equations and the rotation of axes.
Rotated Form: Rotated form refers to the representation of a function or equation in a coordinate system that has been rotated from the standard Cartesian coordinate system. This transformation alters the orientation and position of the original function, allowing for different perspectives and insights into its properties.
Rotation of Axes: Rotation of axes is a mathematical transformation that involves rotating the coordinate system around one or more of the axes. This transformation is often used in analytic geometry and linear algebra to simplify the analysis of geometric shapes and equations.
Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.
Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.
Substitution Formulas: Substitution formulas are mathematical expressions that allow for the replacement of one or more variables in an equation with a specific value or function. These formulas enable the simplification and transformation of complex equations, facilitating the process of solving problems in various mathematical contexts, including the rotation of axes.
Theta: Theta, often represented by the Greek letter θ, is a fundamental mathematical angle that is used extensively in various fields, including trigonometry, geometry, and physics. It is a measure of the rotation or orientation of an object or a coordinate system.
Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It provides the tools and techniques to analyze and solve problems involving angles, lengths, and the relationships between them.
Vertex: The vertex is a critical point in various mathematical functions and geometric shapes. It represents the point of maximum or minimum value, or the point where a curve changes direction. This term is particularly important in the context of quadratic equations, functions, absolute value functions, and conic sections such as the ellipse and parabola.
X-prime: x-prime, or the derivative of x, is a fundamental concept in calculus that represents the rate of change of a function with respect to the independent variable. It is a crucial tool for analyzing and understanding the behavior of functions, particularly in the context of rotation of axes.
Y-prime: Y-prime, often denoted as 'y'', represents the derivative of a function y with respect to its variable, typically x. This concept is essential for understanding rates of change, slopes of curves, and behavior of functions in relation to their inputs. It connects closely to concepts like tangent lines, optimization, and motion, providing insights into how functions vary at specific points.
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Glossary
Glossary
12.5 Conic Sections in Polar Coordinates