MATH04 Pre Calculus - Course Outcome 6: Lesson 1 - Ellipses

MATH04 Pre Calculus - Course Outcome 6: Lesson 1 - Ellipses

Definition of Ellipses

• Ellipse: The locus of a point in a plane which moves such that the sum of its distances from two fixed points (called foci denoted as $f1$ and $f2$) is constant.

  • Notation:

  • Distance 1: $d_1$

  • Distance 2: $d_2$

• Mathematical Representation:

  • For any point $P$ on the ellipse, the relationship can be expressed as:
    $d<em>1 + d</em>2 = d<em>1' + d</em>2'$

Components of Ellipses

• Foci: Points $f1$ and $f2$ that are the fixed points from which distances to points on the ellipse are measured.

• Major Axis: The segment cut by the ellipse on the line containing the foci; the longest diameter of the ellipse.

• Minor Axis: The segment cut by the ellipse that is perpendicular to the major axis.

Essential Terminology

• Vertices: The endpoints of the diameter through the foci.

• Latus Rectum: A segment cut by the ellipse that passes through the foci, utilized in defining the ellipse's shape.

• Eccentricity ($e$): A measure of the degree of flatness of an ellipse, calculated as:
  $e = \frac{c}{a}$
  where $c$ is the distance from the center to each focus and $a$ is the semi-major axis length.

Standard Equations of Ellipses

• Horizontal Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

  • Where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis.

• Vertical Ellipse:
  $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

• Pythagorean Relations: For an ellipse, these relationships define how the axes are related: $c^2 = a^2 - b^2$ where:

  • $c$ = distance from center to foci

  • $a$ = semi-major axis length

  • $b$ = semi-minor axis length

Summary Table of Ellipse Properties

• Center: $(h, k)$

• Foci: $f1$, $f2$

• Length of Major Axis: $2a$

• Vertices of Major Axis: $(h \pm a, k)$

• Length of Minor Axis: $2b$

• Vertices of Minor Axis: $(h, k \pm b)$

• Length of Latus Rectum: $\frac{2b^2}{a}$

• Vertices of Latus Rectum: $\left(h \pm c, k \pm \frac{b^2}{a}\right)$

• Eccentricity: $e = \frac{c}{a}$

• Area: $A = \pi ab$

Real-life Applications of Ellipses

• Orbits: Describe the orbits of planets, comets, moons, and satellites, which are elliptical in shape.

• Design: Shape of boats and wings of some airplanes.

• Medical Devices: Such as lithotripters that utilize elliptical reflectors to break up kidney stones.

• Architectural Structures: Some buildings are designed with elliptical domes, known as whispering chambers.

Examples and Exercises

Example 1: Properties of an Ellipse

• Given the ellipse: Identify properties including center, lengths of major and minor axes, and other key metrics.

  • Center $(h, k)$: $C(2, 1)$

  • Length of Major Axis = $10$ units

  • Length of Minor Axis = $8$ units

  • Semi-major axis $a = 5$ units, semi-minor axis $b = 4$ units.

• Foci Calculation: Using Pythagorean relation: If $a = 5$ and $b = 4$, then $c = \sqrt{a^2 - b^2} = \sqrt{25 - 16} = 3.$

  • Foci: $f1(-1, 1)$ and $f2(5, 1)$.

• Eccentricity: $e = \frac{c}{a} = \frac{3}{5} = 0.6.$

  • Area: $A = \pi ab = \pi(5)(4) = 20\pi.$

Example 2: Equation of Ellipse

• Find the equation of an ellipse given specific conditions.

  • Center is at $(0, 0)$, foci at $(-\sqrt{3}, 0)$ and $(+\sqrt{3}, 0)$ for a horizontal ellipse.

Example 3: Satellite Orbit

• Given: A satellite orbits the earth in an elliptical path with eccentricity $0.80$ and a semi-major axis of length $20,000$ km.

  • Find: Maximum altitude (apogee) and corresponding dimensions based on the center located at one focus of the ellipse.

Example 4: Semi-Elliptical Arch

• A semi-ellipse with a span of 45m and a maximum height of 12m with vertical supports.

  • Find: The height of the vertical supports which are equidistant from the ends of the arc.

Exercises

Part A: Standard Form Conversion

1. Reduce the equation to standard form. Determine the center, vertices, ends of the minor axis, foci, latera recta, length of major and minor axis.

Part B: Equation Finding

1. Find the equation of an ellipse satisfying specific conditions (major axis lengths, focal points, etc.).

Conclusion

• Understanding the characteristics and properties of ellipses is crucial in both mathematical theory and practical applications. Mastery of these concepts lays the foundation for further study in conic sections and their geometric attributes.