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Conic Sections definitions

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  • Conic Section
    A plane curve formed by slicing a cone with a plane, resulting in shapes like circles, ellipses, parabolas, and hyperbolas.
  • Circle
    A set of points equidistant from a fixed center, represented by an equation with equal squared terms for x and y.
  • Ellipse
    An oval-shaped curve with two axes of symmetry, defined by two distances from the center: the semi-major and semi-minor axes.
  • Parabola
    A U-shaped curve where each point is equidistant from a fixed point (focus) and a fixed line (directrix).
  • Hyperbola
    A curve with two branches opening away from each other, defined by the difference of distances to two foci being constant.
  • Vertex
    A key point on a conic section, such as the maximum or minimum point of a parabola or the closest point to the center in a hyperbola.
  • Focus
    A special point used to define conic sections, where distances from this point relate to the curve's symmetry and properties.
  • Directrix
    A fixed line used in the definition of a parabola, equidistant from the vertex as the focus but in the opposite direction.
  • Asymptote
    A straight line that a curve approaches but never touches, crucial for graphing hyperbolas.
  • Major Axis
    The longest diameter of an ellipse, passing through its center and both foci.
  • Minor Axis
    The shortest diameter of an ellipse, perpendicular to the major axis at the center.
  • Standard Form
    A simplified equation of a conic section, making it easy to identify key features like center, axes, and orientation.
  • General Form
    An expanded equation of a conic section, often requiring algebraic manipulation to reveal geometric properties.
  • Center
    The midpoint of symmetry for circles, ellipses, and hyperbolas, often denoted by coordinates (h, k).
  • Axis of Symmetry
    A line that divides a conic section into two mirror-image halves, crucial for graphing and analysis.