Chapter 10: Conic Sections Study Notes

10.1 Introduction

• Chapter Focus: Study of curves including circles, ellipses, parabolas, and hyperbolas.

• Origin of Names: The terms "parabola" and "hyperbola" were introduced by Apollonius.

• Definition: Curves known as conic sections or conics; formed by the intersection of a plane with a double napped right circular cone.

• Applications:

  • Planetary motion

  • Design of telescopes and antennas

  • Reflectors in flashlights and automobile headlights

• Objective: Explore intersection of plane with double-napped cone to derive various conic sections.

10.2 Sections of a Cone

• Setup:

  • Let line l be fixed vertically.

  • Line m intersects line l at a fixed point V with angle α.

  • Rotating line m around line l creates a double-napped cone.

• Definitions:

  • Vertex (V): Point where the two nappes meet.

  • Axis (l): The fixed vertical line.

  • Generator (m): The rotating line that forms the cone.

  • Nappes: Two parts of the cone created by the vertex.

• Intersection Properties:

  • A plane intersecting a cone creates a conic section.

  • Different cases arise based on the angle β (angle of the plane with the vertical axis).

10.2.1 Types of Conic Sections

1. Circle:

   • Condition: When β = 90° (horizontal intersection).

   • Example: Circle shape.

2. Ellipse:

   • Condition: When α < β < 90°.

   • Example: Elongated circle.

3. Parabola:

   • Condition: When β = α.

   • Example: U-shaped curve.

4. Hyperbola:

   • Condition: When 0 ≤ β < α.

   • Example: Two separate curves that open away from each other.

10.2.2 Degenerated Conic Sections

• Intersection at the Vertex:

1. Point: (α < β ≤ 90°)

2. Straight Line: (β = α); this is the degenerate case of a parabola.

3. Pair of Intersecting Lines: (0 ≤ β < α); this is the degenerated case of a hyperbola.

10.3 Circle

• Definition 1: A circle is defined as the set of all points in a plane that are equidistant from a fixed point (the center of the circle).

• Radius: Distance from the center to any point on the circle.

• Figure Representation: Figure 10.11 illustrates the aforementioned properties.

10.3.1 Equation of the Circle

• Simplest form: When the center is at the origin (0, 0).

• General form: When the center of the circle is at (h, k) and radius r, the equation is given as:
  $(x - h)^2 + (y - k)^2 = r^2$

• Examples:

  • Example 1: Circle with center (0,0) radius r.

  • Equation: $x^2 + y^2 = r^2$

  • Example 2: Circle with center (–3, 2) and radius 4.

  • Equation: $(x + 3)^2 + (y - 2)^2 = 16$

  • Example 3: Find center and radius of $x^2 + y^2 + 8x + 10y - 8 = 0$

  • Completing the square leads to: $(x + 4)^2 + (y + 5)^2 = 49$

  • Result: Center is (–4, –5) and radius is 7.

  • Example 4: Circle passing through points (2, –2) and (3, 4) with center on the line x + y = 2.

  • Setup of equations based on circle definition results in (x – 0.7)² + (y – 1.3)² = 12.58.

10.4 Parabola

• Definition 2: A parabola is the set of all points equidistant from a fixed line (the directrix) and a fixed point (the focus) in the plane.

• Axis of Parabola: Line through focus perpendicular to the directrix.

• Vertex: Intersection of the parabola with its axis.

• Visual Representation: Figure 10.13 shows focus and directrix.

10.4.1 Standard Equations of Parabola

• Focus at (a, 0) where a > 0; directrix at x = –a.

• Calculation for the parabola: From distance properties, we derive:

  • $y^2 = 4ax$ where the parabola opens to the right.

  • Other orientations lead to:

  • $y^2 = -4ax$ (opens left)

  • $x^2 = 4ay$ (opens up)

  • $x^2 = -4ay$ (opens down)

• Observations:

  • Symmetry and orientation of parabolas depend on coefficients being positive or negative.

10.4.2 Latus Rectum

• Definition 3: Latus rectum is the line segment through a focus, perpendicular to the axis of the parabola. Points at the end intersect the parabola.

• Length of the Latus Rectum: For the parabola $y^2 = 4ax$, length is 4a.

• Examples:

  • Find coordinates of focus, axis, directrix, and length of latus rectum for $y^2 = 8x$. Result provides all necessary information for the parabola.

10.5 Ellipse

• Definition 4: An ellipse is defined as the set of points in a plane where the sum of distances from two fixed points (foci) remains constant.

• Properties:

  • Constant sum of distances > Distance between foci.

  • Points at the ends of the major axis are vertices.

• Visual Representation: Includes Configurations of the ellipse and associated lines through the foci.

10.5.1 Relationship Between Axes

• Establish relationships for semi-major axis (2a), semi-minor axis (2b), and distance of foci from the center (c).

• Relation defines:

  • $a^2 = b^2 + c^2$ and therefore $c = \ rac{a^2 - b^2}{a}$

10.5.2 Eccentricity

• Definition 5: Eccentricity (e) of an ellipse related to distance ratios: $e = \ rac{c}{a}$

• Focus: Distance of the focus defined as ae from the center.

10.5.3 Standard Equations of Ellipse

• Simplest equation: Center at origin aligned with axes.

• Two orientations:

  1. Foci on x-axis: $\ rac{x^2}{a^2} + \ rac{y^2}{b^2} = 1$

  2. Foci on y-axis: $\ rac{y^2}{a^2} + \ rac{x^2}{b^2} = 1$

• Foci Properties: Always lie along major axis, and ellipse symmetric with respect to both axes.

10.5.4 Latus Rectum

• Definition 6: Latus rectum for ellipse perpendicular to the major axis and intersects the ellipse at endpoints.

• Length: $l = \ rac{2b^2}{a}$

• Examples: Provide identification of essential properties for given equations.

10.6 Hyperbola

• Definition 7: Hyperbola set is the difference of distances from two points being constant (Foci).

• Geometric representation includes axes and transverse points.

10.6.1 Eccentricity

• Definition 8: Eccentricity involves ratio $e = \ rac{c}{a}$ where c ≥ a.

10.6.2 Standard Equation of Hyperbola

• Simplest centered at origin with orientations: $\ rac{x^2}{a^2} - \ rac{y^2}{b^2} = 1$ and $\ rac{y^2}{a^2} - \ rac{x^2}{b^2} = 1$

10.6.3 Latus Rectum

• Definition 9: Length of latus rectum in hyperbola is $l = \ rac{2b^2}{a}$

• Examples: Various examples calculated and discussed.

Miscellaneous Examples

• Example problems covering rays, rod configurations, and beam support analyzed through geometric attributes leading to conic sections.

Summary

• Circle: Set of equidistant points from a fixed point. - Circle Equation: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: Equidistant points from directrix and a focus. - Standard Equation: $y^2 = 4ax$

• Latus Rectum: Length in parabola is 4a.

• Ellipse: Set where sum of distances from two foci is constant.

• Ellipse Equation: $\ rac{x^2}{a^2} + \ rac{y^2}{b^2} = 1$

• Hyperbola: Set where the difference of distances from two foci is constant.

• Hyperbola Equation: $\ rac{x^2}{a^2} - \ rac{y^2}{b^2} = 1$

• Latus Rectum: For hyperbola $l = \ rac{2b^2}{a}$

• Eccentricities defined for both ellipse and hyperbola indicating distance ratios.