📘Intermediate Algebra Unit 11 – Conics

What Are Conics?

• Conics are curves formed by the intersection of a plane with a double cone
• Derived from the Greek word "konos" meaning cone
• Studied in mathematics, particularly in analytic geometry
• Conics include circles, ellipses, parabolas, and hyperbolas
• Defined by a general second-degree equation in two variables, $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$
• Conics have various geometric properties and applications in fields such as physics, engineering, and astronomy
• The type of conic section depends on the angle at which the plane intersects the double cone
  • If the plane is parallel to the base of the cone, the intersection is a circle
  • If the plane is tilted, the intersection can be an ellipse, parabola, or hyperbola

Types of Conic Sections

• There are four main types of conic sections: circles, ellipses, parabolas, and hyperbolas
• Circles are formed when a plane intersects a cone perpendicular to its axis
• Ellipses are formed when a plane intersects a cone at an angle less than the angle between the cone's axis and its generator
  • An ellipse has two foci and the sum of the distances from any point on the ellipse to the foci is constant
• Parabolas are formed when a plane intersects a cone parallel to one of its generators
  • A parabola has a single focus and a directrix, and any point on the parabola is equidistant from the focus and the directrix
• Hyperbolas are formed when a plane intersects both nappes of a double cone
  • A hyperbola has two branches and two foci, and the difference of the distances from any point on the hyperbola to the foci is constant
• Degenerate cases of conics include a point, a line, or a pair of intersecting lines

The Circle: Basics and Equations

• A circle is a closed curve in which all points are equidistant from a fixed point called the center
• The distance from the center to any point on the circle is called the radius
• The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$
  • If the center is at the origin $(0, 0)$, the equation simplifies to $x^2 + y^2 = r^2$
• The general form of the equation of a circle is $x^2 + y^2 + Dx + Ey + F = 0$, where $D$, $E$, and $F$ are constants
• To find the center and radius of a circle from its general form, complete the square for both $x$ and $y$ terms
• Circles have various properties, such as:
  • The diameter is twice the radius and passes through the center
  • Chords are line segments connecting any two points on the circle
  • Tangent lines touch the circle at exactly one point and are perpendicular to the radius at that point

Ellipses: Definition and Properties

• An ellipse is a closed curve defined as the set of points in a plane such that the sum of the distances from any point on the curve to two fixed points (foci) is constant
• The standard form of the equation of an ellipse with center $(h, k)$ and major and minor axis lengths $a$ and $b$ is:
  • $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ (horizontal major axis)
  • $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ (vertical major axis)
• The foci of an ellipse lie on its major axis and are equidistant from the center
  • The distance between the foci is $2c$, where $c^2 = a^2 - b^2$
• The eccentricity of an ellipse, denoted by $e$, is the ratio of the distance between the foci to the length of the major axis ($e = \frac{c}{a}$)
  • Eccentricity measures how much an ellipse deviates from a circle (e = 0 for a circle)
• Ellipses have two vertices (points where the major axis intersects the ellipse) and two co-vertices (points where the minor axis intersects the ellipse)
• The latus rectum of an ellipse is a chord parallel to the minor axis and passing through a focus, with length $\frac{2b^2}{a}$

Parabolas: Vertex Form and Applications

• A parabola is a symmetrical open curve formed by the intersection of a cone with a plane parallel to one of its generators
• Parabolas have a single focus and a directrix (a line perpendicular to the axis of symmetry)
  • Any point on the parabola is equidistant from the focus and the directrix
• The vertex form of the equation of a parabola with vertex $(h, k)$, axis of symmetry parallel to the y-axis, and focal length $p$ is:
  • $(y - k) = \frac{1}{4p}(x - h)^2$ (opens upward)
  • $(y - k) = -\frac{1}{4p}(x - h)^2$ (opens downward)
• For parabolas with axis of symmetry parallel to the x-axis, the vertex form is:
  • $(x - h) = \frac{1}{4p}(y - k)^2$ (opens rightward)
  • $(x - h) = -\frac{1}{4p}(y - k)^2$ (opens leftward)
• The focus of a parabola is located $p$ units from the vertex along the axis of symmetry
• Parabolas have many applications, such as:
  • Projectile motion (trajectory of objects under the influence of gravity)
  • Satellite dishes and reflective telescopes (parabolic reflectors)
  • Headlights and flashlights (parabolic reflectors)
  • Suspension bridges (parabolic cables)

Hyperbolas: Characteristics and Graphs

• A hyperbola is an open curve with two branches, formed by the intersection of a double cone with a plane perpendicular to the cone's axis
• Hyperbolas have two foci and two vertices (points where the hyperbola intersects its transverse axis)
  • The difference of the distances from any point on the hyperbola to the foci is constant
• The standard form of the equation of a hyperbola with center $(h, k)$ and transverse and conjugate axis lengths $a$ and $b$ is:
  • $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (horizontal transverse axis)
  • $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ (vertical transverse axis)
• The foci of a hyperbola lie on its transverse axis and are equidistant from the center
  • The distance between the foci is $2c$, where $c^2 = a^2 + b^2$
• The eccentricity of a hyperbola, denoted by $e$, is the ratio of the distance between the foci to the length of the transverse axis ($e = \frac{c}{a}$)
  • Eccentricity is always greater than 1 for hyperbolas
• Hyperbolas have two asymptotes (lines that the branches approach but never touch)
  • The equations of the asymptotes are $y = \pm\frac{b}{a}(x - h) + k$
• Applications of hyperbolas include:
  • Locating the position of an object using time difference of arrival (TDOA) in navigation systems
  • Modeling the paths of comets and other celestial bodies in astronomy
  • Designing cooling towers for power plants (hyperboloid structures)

General Form of Conic Equations

• The general form of a conic equation is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $A$, $B$, $C$, $D$, $E$, and $F$ are constants, and $A$, $B$, and $C$ are not all zero
• The type of conic section can be determined by the discriminant $B^2 - 4AC$:
  • If $B^2 - 4AC < 0$, the conic is an ellipse
  • If $B^2 - 4AC = 0$, the conic is a parabola
  • If $B^2 - 4AC > 0$, the conic is a hyperbola
  • If $A = C$ and $B = 0$, the conic is a circle
• To identify the center, vertices, and other properties of a conic from its general form, the equation can be transformed into standard form by:
  • Completing the square for $x$ and $y$ terms
  • Rotating the coordinate system to eliminate the $xy$ term (if $B \neq 0$)
• The general form can also represent degenerate cases, such as:
  • A single point (if the conic equation factors into two identical linear factors)
  • A pair of parallel lines (if the conic equation factors into two distinct linear factors)
  • A pair of intersecting lines (if the conic equation factors into two linear factors)
• Understanding the general form of conic equations allows for the analysis and classification of conics in various orientations and positions in the coordinate plane

Real-World Applications of Conics

• Conics have numerous real-world applications in various fields, such as physics, engineering, and astronomy
• In architecture, elliptical and parabolic arches are used in bridges and buildings for their strength and aesthetic appeal
  • The Gateway Arch in St. Louis, USA, is a famous example of a catenary arch, which closely resembles a parabola
• In optics, parabolic mirrors are used in telescopes, satellite dishes, and solar cookers to focus light or electromagnetic waves
  • The Hubble Space Telescope uses a parabolic mirror to collect and focus light from distant galaxies
• Elliptical gears are used in machines to convert rotary motion into reciprocating motion or vice versa
  • Elliptical trainers, a type of exercise equipment, use elliptical gears to simulate a running motion with reduced impact on joints
• In physics, the paths of projectiles under the influence of gravity follow a parabolic trajectory
  • This principle is applied in sports such as basketball, golf, and archery to optimize the path of the ball or arrow
• Hyperbolic geometry, which is based on hyperbolas, has applications in special relativity and the study of spacetime
  • The Minkowski diagram, used to visualize spacetime events, is an example of a hyperbolic geometry concept
• In astronomy, the orbits of comets and other celestial bodies around the sun can be modeled as ellipses or hyperbolas
  • Halley's Comet, which orbits the sun every 75-76 years, follows an elliptical path
• Understanding the properties and applications of conics enables professionals in various fields to solve problems, design efficient systems, and make new discoveries