Lecture Notes: Review of Conic Sections

Introduction

• Conic Sections: Parabolas, ellipses, and hyperbolas are called conic sections or conics. They result from intersecting a cone with a plane.

Parabolas

• Definition: Set of points equidistant from a fixed point (focus) and a fixed line (directrix).
• Vertex: The point halfway between focus and directrix on the parabola.
• Axis: Line through the focus perpendicular to the directrix.
• Equation: Vertex at origin, directrix parallel to the y-axis: (x^2 = 4py).
  • Opens upward if (p > 0), downward if (p < 0).
  • Reflecting telescopes, suspension bridges, etc., are practical uses.
• Example: Find focus and directrix of (x^2 = 4py).
  • Result: Focus ((0,1)), directrix (y = -1).

Ellipses

• Definition: Set of points where the sum of distances from two fixed points (foci) is constant.
• Equation: (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1), with foci on x-axis at ((c,0), (-c,0)).
  • (a^2 = c^2 + b^2).
• Properties: Symmetric about both axes.
• Example: Sketch and locate foci for (\frac{x^2}{16} + \frac{y^2}{9} = 1).
  • Result: Foci at ((\pm\sqrt{7},0)).

Hyperbolas

• Definition: Set of points where the difference of distances from two fixed points (foci) is constant.
• Equation: (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1) or (\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1).
  • (c^2 = a^2 + b^2).
• Properties: Asymptotes are (y = \pm\frac{b}{a}x).
• Example: Find foci and asymptotes for (\frac{x^2}{16} - \frac{y^2}{9} = 1).
  • Result: Foci at ((\pm\sqrt{25},0)), asymptotes (y=\pm\frac{3}{4}x).

Shifted Conics

• Shifting: Replace x and y with ((x - h)) and ((y - k)) in standard equations.
• Example: Ellipse ((x-2)^2/9 + (y-1)^2/4 = 1) with center ((2,1)).

Practical Applications

• Parabolic Reflection: Used in headlights, telescopes.
• Elliptical Reflection: Used in lithotripsy for kidney stones.
• Hyperbolic Navigation: Used in navigation systems (e.g., LORAN).

Exercises and Applications

• The lecture included numerous example problems, exercises, and applications related to each type of conic section.

Reflection Properties

• Parabolas, ellipses, and hyperbolas have unique reflection properties that have practical applications in various fields such as astronomy, medicine, and navigation.