pre cal

Intersecting Planes and Conic Sections

• A conic section is formed by the intersection of a double-napped cone and a plane.
• The angle and position of the intersecting plane determine the type of conic section produced (circle, ellipse, parabola, or hyperbola).

Eccentricity of Conic Sections

• The eccentricity (e) measures how much a conic section deviates from being circular.
• Eccentricity values:
  • e = 0 corresponds to a circle.
  • 0 < e < 1 corresponds to an ellipse.
  • e = 1 corresponds to a parabola.
  • e > 1 corresponds to a hyperbola.

General Form of a Conic Section

• The general quadratic equation of conic sections is given by:
  • Ax² + Bxy + Cy² + Dx + Ey + F = 0.
• A, B, C, D, E, and F are constants.

Quadratic Terms of Conic Sections

• The quadratic terms refer to Ax², Bxy, and Cy² in the general equation.
• The coefficients A, B, and C play crucial roles in identifying the type of conic section represented.

Discriminant Formula

• The discriminant for conic sections is defined as:
  • D = B² - 4AC.
• This formula helps classify the conic section based on the values of A, B, and C.

Classification of Conic Sections Using Discriminant

• B² - 4AC < 0 indicates an ellipse or a circle.
• B² - 4AC = 0 indicates a parabola.
• B² - 4AC > 0 indicates a hyperbola.

Identifying Ellipses and Circles

• To determine whether a conic section is an ellipse or circle:
  • Check the coefficients A and C:
    • If A = C and B = 0, the conic is a circle.
    • If A ≠ C and B = 0, the conic is an ellipse.