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Conic Sections in Precalculus

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  • What is a conic section?
    A conic section is the curve formed by the intersection of a plane and a double-napped cone. It includes parabolas, ellipses, and hyperbolas.
  • Define a parabola.
    A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.
  • Standard form of a vertical parabola equation.
    The standard form is \((x-h)^2=4p(y-k)\), where (h,k) is the vertex and p is the distance from vertex to focus.
  • What is the focus of a parabola?
    The focus is a fixed point inside the parabola used to define it; every point on the parabola is equidistant from the focus and the directrix.
  • Define an ellipse.
    An ellipse is the set of all points where the sum of the distances to two fixed points called foci is constant.
  • Standard form of an ellipse with horizontal major axis.
    The equation is \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\), where a > b and (h,k) is the center.
  • What is the relationship between a, b, and c in an ellipse?
    They satisfy \(c^2=a^2-b^2\), where c is the distance from center to each focus.
  • Define a hyperbola.
    A hyperbola is the set of all points where the absolute difference of the distances to two fixed points called foci is constant.
  • Standard form of a hyperbola with horizontal transverse axis.
    The equation is \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\), where (h,k) is the center.
  • Relationship between a, b, and c in a hyperbola.
    They satisfy \(c^2=a^2+b^2\), where c is the distance from center to each focus.
  • What is the directrix of a parabola?
    The directrix is a fixed line used to define a parabola; every point on the parabola is equidistant from the focus and the directrix.
  • How to find the vertices of an ellipse?
    Vertices lie along the major axis at a distance a from the center (h,k).
  • How to find the vertices of a hyperbola?
    Vertices lie along the transverse axis at a distance a from the center (h,k).
  • What is the eccentricity of a conic section?
    Eccentricity e measures how much a conic deviates from being circular; for ellipses, \(0e=1, and for hyperbolas, \(e>1\).
  • Formula for eccentricity of an ellipse.
    Eccentricity is \(e=\frac{c}{a}\), where c is the focal distance and a is the semi-major axis.
  • Formula for eccentricity of a hyperbola.
    Eccentricity is \(e=\frac{c}{a}\), where c is the focal distance and a is the distance from center to vertex.
  • What is the axis of symmetry in a parabola?
    The axis of symmetry is the line through the vertex and focus that divides the parabola into two mirror images.
  • How to identify the conic from the general quadratic equation?
    Use the discriminant \(B^2-4AC\): if zero, parabola; if less than zero, ellipse; if greater than zero, hyperbola.
  • What is the latus rectum of a parabola?
    The latus rectum is the chord through the focus perpendicular to the axis of symmetry; its length is \(|4p|\).
  • How to find the foci of an ellipse?
    Foci are located along the major axis at a distance c from the center, where \(c=\sqrt{a^2-b^2}\).