MATH04 Pre Calculus Ellipses Study Notes

Ellipse: The locus of a point in a plane that moves so that the sum of its distances from two fixed points (known as foci, denoted as $F1$ and $F2$) is constant.
- Mathematically expressed as:
  $d<em>1 + d</em>2 = k$
- Where $d1$ and $d2$ are the distances from a point on the ellipse to the foci, and $k$ is a constant greater than the distance between the foci.

Foci: The two fixed points $F1(hc, k)$ and $F2(h+c, k)$ through which the ellipse is defined.
Major Axis: The longest diameter of the ellipse, passing through the foci.
Minor Axis: The diameter that is perpendicular to the major axis.

Vertices: The endpoints of the major axis; points where the ellipse intersects the major axis.
Latus Rectum: A segment perpendicular to the major axis that passes through a focus.
Eccentricity ($e$): A measure of how much the ellipse deviates from being circular. Calculated as:
$e = rac{c}{a}$
where $c$ is the distance from the center to a focus, and $a$ is the semi-major axis length.

Horizontal Ellipse: If the major axis is horizontal, the standard form is:
$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
- Where $(h, k)$ is the center.
- Here, $a$ is the semi-major axis and $b$ is the semi-minor axis.
Vertical Ellipse: If the major axis is vertical, the standard form is:
$\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$

Celestial Orbits: Orbits of planets, comets, moons, and satellites are elliptical.
Design: The shape of boats and certain airplane wings.
Medical Devices: Lithotripters utilize elliptical reflectors to target kidney stones.
Architecture: Whispering chambers are built with elliptical domes to enhance acoustics.

Given the ellipse described by the equation:
$16x^2 + 25y^2 - 64x - 50y - 311 = 0$

Completing the Square: Rearranging the equation by completing the square.
- $16[x^2 - 4x] + 25[y^2 - 2y] = 311$
- Further processing yields the standard form of the ellipse.
Find Semi-Major and Semi-Minor Axes ($a$ and $b$):
- Use the relationships established in the quadratic forms to extract lengths and center:
- Center calculated at $(h, k) = (2, 1)$.
Calculate Lengths:
- Length of Major Axis = $2a$ where $a = 5$ → $Length = 10$ units.
- Length of Minor Axis = $2b$ where $b = 4$ → $Length = 8$ units.
Calculate Latus Rectum:
$Length = \frac{2b^2}{a} = \frac{2(4)^2}{5} = \frac{32}{5}$
Vertices and Foci:
- Vertices: $(2 - 5, 1), (2 + 5, 1)$ yields $(−3, 1)$ and $(7, 1)$.
- Foci positions derived through $c= ext{sqrt(a}^2 - ext{b}^2$:
  - Calculated as $(2-c, 1), (2+c, 1)$ where $c=3$.
Eccentricity and Area:
- Using $e = \frac{c}{a}$ yields values reflecting the ellipse's characteristics.

Scenario: Given specific focal points, lengths, and orientations; derive the equation that fits these conditions, ensuring to apply principles discussed above regarding center, eccentricity, and alignment.

A satellite with a focus on the Earth at one point, calculate the maximum altitude using semi-major axis given the eccentricity, determining apogee and perigee based on these parameters.

For an arch modeled as a semi-ellipse, calculate the heights of vertical supports based on span and maximum height, applying the properties of ellipses in practical architectural design scenarios.

Example problems where students are asked to rewrite provided equations into standard ellipse form and deduce properties from them.

Exercise problems where students derive the equation of an ellipse given specific geometric conditions and constraints.

Understanding ellipses is vital for various fields such as mathematics, physics, architecture, and engineering, providing insights into the behavior of objects moving in elliptical paths and the impacts of those shapes in practical scenarios.