Ellipses and hyperbolas are key players in the conic sections family. These curves have unique properties that make them useful in various fields. From planetary orbits to architectural designs, they're everywhere!

Understanding ellipses and hyperbolas is crucial for grasping conic sections. We'll look at their equations, how to graph them, and real-world applications. Get ready to see these shapes in a whole new light!

Ellipses and Hyperbolas: Key Features

Ellipses

• Ellipse is a conic section formed by the intersection of a plane with a double cone resulting in a closed curve
• Standard form equation of an ellipse with center at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes
• Foci of an ellipse are two points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant
  • Foci lie on the major axis, and their coordinates are $(\pm c, 0)$ in the standard form, where $c^2 = a^2 - b^2$
• Eccentricity of an ellipse, denoted by $e$, is the ratio of the distance from the center to a focus and the length of the semi-major axis
  • Measure of how elongated the ellipse is, with $0 < e < 1$

Hyperbolas

• Hyperbola is a conic section formed by the intersection of a plane with a double cone resulting in two separate unbounded curves called branches
• Standard form equation of a hyperbola with center at the origin is:
  • Horizontal transverse axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
  • Vertical transverse axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
  • $a$ and $b$ are the lengths of the transverse and conjugate axes
• Foci of a hyperbola are two points, one on each branch, such that the difference of the distances from any point on the hyperbola to the foci is constant
  • Foci lie on the transverse axis, and their coordinates are $(\pm c, 0)$ or $(0, \pm c)$ in the standard form, where $c^2 = a^2 + b^2$
• Eccentricity of a hyperbola, denoted by $e$, is the ratio of the distance from the center to a focus and the length of the transverse axis
  • Always greater than 1, i.e., $e > 1$
• Hyperbolas have two asymptotes, which are lines that the branches approach but never touch
  • Equations of the asymptotes are $y = \pm\frac{b}{a}x$ for a hyperbola with a horizontal transverse axis and $y = \pm\frac{a}{b}x$ for a hyperbola with a vertical transverse axis

Graphing Ellipses and Hyperbolas

Graphing Ellipses

• To graph an ellipse in standard form, follow these steps:
  1. Identify the center $(h, k)$ and the lengths of the semi-major and semi-minor axes ($a$ and $b$)
  2. Plot the center and the vertices $(h \pm a, k)$ and $(h, k \pm b)$
  3. Sketch the ellipse using these points as a guide
• General form equation of an ellipse is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $(h, k)$ is the center of the ellipse

Graphing Hyperbolas

• To graph a hyperbola in standard form, follow these steps:
  1. Identify the center $(h, k)$, the lengths of the transverse and conjugate axes ($a$ and $b$), and the orientation (horizontal or vertical)
  2. Plot the center and the vertices $(h \pm a, k)$ for a horizontal hyperbola or $(h, k \pm a)$ for a vertical hyperbola
  3. Plot the asymptotes using their equations
  4. Sketch the hyperbola using these points and asymptotes as a guide
• General form equations of a hyperbola are:
  • Horizontal: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
  • Vertical: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$
  • $(h, k)$ is the center of the hyperbola
• To convert an equation from general form to standard form:
  1. Complete the square for both the $x$ and $y$ terms
  2. Isolate the constant term on the right side of the equation

Equations of Ellipses and Hyperbolas

Finding Equations of Ellipses

• To find the equation of an ellipse given the center, vertices, and co-vertices (or endpoints of the minor axis):
  1. Determine the lengths of the semi-major and semi-minor axes ($a$ and $b$)
  2. Substitute the values of $h$, $k$, $a$, and $b$ into the general form equation
• To find the equation of an ellipse given the foci and a point on the ellipse:
  1. Use the definition of an ellipse: the sum of the distances from any point on the ellipse to the foci is constant
  2. Set up an equation using this property and solve for the constant sum
  3. Use the coordinates of the foci and the constant sum to write the equation in standard form

Finding Equations of Hyperbolas

• To find the equation of a hyperbola given the center, vertices, and a point on the hyperbola:
  1. Determine the lengths of the transverse and conjugate axes ($a$ and $b$)
  2. Substitute the values of $h$, $k$, $a$, and $b$ into the general form equation
• To find the equation of a hyperbola given the foci and a point on the hyperbola:
  1. Use the definition of a hyperbola: the difference of the distances from any point on the hyperbola to the foci is constant
  2. Set up an equation using this property and solve for the constant difference
  3. Use the coordinates of the foci and the constant difference to write the equation in standard form
• To find the equation of a hyperbola given the asymptotes and a point on the hyperbola:
  1. Find the center of the hyperbola by intersecting the asymptotes
  2. Use the slope of the asymptotes and the given point to determine the lengths of the transverse and conjugate axes ($a$ and $b$)
  3. Substitute the values of $h$, $k$, $a$, and $b$ into the general form equation

Applications of Ellipses and Hyperbolas

Ellipses in Real-World Scenarios

• Ellipses can be used to model the orbits of planets and satellites around celestial bodies, as well as the paths of objects in motion under the influence of gravity or other forces
  • Kepler's first law of planetary motion states that the orbits of planets around the sun are elliptical, with the sun at one focus (Earth's orbit, Mars' orbit)
  • The eccentricity of a planetary orbit determines how elongated the ellipse is, with a higher eccentricity indicating a more elongated orbit (Mercury's orbit, Halley's comet)
• In architecture and engineering, elliptical shapes are used in the design of structures such as arches and domes to optimize strength, stability, and efficiency (Oval Office in the White House, Capitol Building dome)
• Ellipses are used in the design of reflective surfaces for telescopes and other optical devices to focus electromagnetic waves (Hubble Space Telescope's primary mirror, reflecting telescope designs)

Hyperbolas in Real-World Scenarios

• Hyperbolas can be used to model the paths of objects moving at speeds greater than the speed of sound (supersonic motion), such as the shock waves created by explosions or supersonic aircraft
  • The Mach cone is a conical shock wave that forms around an object moving at supersonic speeds, with the object at the vertex of the cone and the edges of the cone forming a hyperbola (sonic boom, bullet shock waves)
  • The angle between the asymptotes of the hyperbola is determined by the Mach number, which is the ratio of the object's speed to the speed of sound in the medium (Concorde supersonic jet, SR-71 Blackbird)
• In architecture and engineering, hyperbolic shapes are used in the design of structures such as cooling towers to optimize efficiency and natural air circulation
  • The Gateway Arch in St. Louis, Missouri, is an inverted catenary curve, which is closely approximated by a hyperbolic cosine function
  • The shape of a hyperbolic cooling tower allows for efficient heat dissipation and natural air circulation, making it a common design choice for power plants and industrial facilities (nuclear power plant cooling towers, industrial chimneys)
• Hyperbolic mirrors are used in flashlights and searchlights to collimate the light from the bulb into a parallel beam, maximizing the illumination distance and intensity (car headlights, lighthouse Fresnel lenses)