1.4 Conic Sections

Conic sections are fascinating shapes formed when a plane intersects a double cone. They include circles, ellipses, parabolas, and hyperbolas. Each has unique properties and real-world applications, from planetary orbits to satellite dishes.

Understanding conic sections is crucial for grasping more advanced concepts in calculus. These shapes appear in various fields like physics and engineering. Mastering their equations and characteristics will give you a solid foundation for future math and science courses.

Types of Conic Sections

Circular and Elliptical Shapes

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• Circle is a special case of an ellipse where the foci are at the same point (the center)
• Ellipse has two foci and the sum of the distances from any point on the ellipse to the two foci is constant
  • Example: An elliptical orbit of a planet around the sun, where the sun is at one focus

Parabolic Curves

• Parabola is defined as the set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line
  • Example: The path of a projectile launched at an angle to the ground, neglecting air resistance

Hyperbolic Curves

• Hyperbola consists of two separate unbounded curves that open in opposite directions
  • The difference of the distances from any point on the hyperbola to the two foci is constant
  • Example: The path of a spacecraft using a gravitational slingshot around a planet to gain speed

Key Components

Focal Points and Directrix

• Focus (or foci for ellipses and hyperbolas) is a fixed point used in the definition of the conic section
  • Parabola and ellipse have one focus inside the curve, while hyperbola has two foci, one inside each branch
• Directrix is a fixed line used in the definition of a parabola, where all points on the parabola are equidistant from the focus and the directrix

Center and Vertices

• Center is the midpoint of the line segment joining the foci (for ellipses and hyperbolas) or the vertex (for parabolas)
  • Circle's center is the same as its focus
• Vertices are the points where the conic section intersects its axis of symmetry
  • Ellipse has two vertices, parabola has one vertex, and hyperbola has four vertices (two on each branch)

Defining Characteristics

Eccentricity and Conic Forms

• Eccentricity ($e$) measures how much a conic section deviates from being circular
  • Circle: $e = 0$, Ellipse: $0 < e < 1$, Parabola: $e = 1$, Hyperbola: $e > 1$
• Standard form of a conic section is an equation that describes its shape and position relative to the origin
  • Example: Standard form of a circle with center $(h, k)$ and radius $r$: $(x-h)^2 + (y-k)^2 = r^2$
• General form of a conic section is a second-degree equation in two variables, $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where the coefficients determine the type of conic section
  • The general form can be transformed into the standard form by completing the square and rotating/translating the coordinate axes