15.2.1: Resources and Key Concepts

Math 375: Pre-Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Definitions

    • Hyperbola: The set of all points in a plane such that the absolute value of the difference of the distances between the point and two fixed points is a positive constant.
    • Focus (plural: foci): One of the two fixed points used to define a hyperbola.
    • Transverse Axis: The line segment that passes through the center of the hyperbola and has the vertices as its endpoints.
    • Conjugate Axis: The line segment that is perpendicular to the transverse axis, passes through the center, and has the co-vertices as its endpoints.
    • Center (of a hyperbola): The midpoint of both the transverse and conjugate axes.
    • Vertices (of a hyperbola): The endpoints of the transverse axis.
    • Co-vertices (of a hyperbola): The endpoints of the conjugate axis.
    • Central Rectangle: A rectangle centered at the center of the hyperbola with sides that pass through each vertex and co-vertex. The diagonals of this rectangle form the asymptotes of the hyperbola.

    Theorems

    • Standard Form of the Equation of a Hyperbola with Center at the Origin:
      • The standard form for a hyperbola with its transverse axis on the \(x\)-axis is \(\frac{x^2}{v_x^2} - \frac{y^2}{v_y^2} = 1\). The vertices are \((\pm v_x, 0)\), the foci are \((\pm c, 0)\) where \(c^2 = v_x^2 + v_y^2\), and the asymptotes are \(y = \pm \frac{v_y}{v_x}x\).
      • The standard form for a hyperbola with its transverse axis on the \(y\)-axis is \(\frac{y^2}{v_y^2} - \frac{x^2}{v_x^2} = 1\). The vertices are \((0, \pm v_y)\), the foci are \((0, \pm c)\) where \(c^2 = v_x^2 + v_y^2\), and the asymptotes are \(y = \pm \frac{v_y}{v_x}x\).
    • Standard Form of the Equation of a Hyperbola with Center at \((h,k)\):
      • The standard form for a hyperbola with its transverse axis parallel to the \(x\)-axis is \(\frac{(x-h)^2}{v_x^2} - \frac{(y-k)^2}{v_y^2} = 1\). The equations of the asymptotes are \(y-k = \pm \frac{v_y}{v_x}(x-h)\).
      • The standard form for a hyperbola with its transverse axis parallel to the \(y\)-axis is \(\frac{(y-k)^2}{v_y^2} - \frac{(x-h)^2}{v_x^2} = 1\). The equations of the asymptotes are \(y-k = \pm \frac{v_y}{v_x}(x-h)\).

    Common Mistakes

    • Confusing the Foci Equation with that of an Ellipse: For a hyperbola, the relationship is \(c^2 = v_x^2 + v_y^2\). This is different from the ellipse, which uses subtraction.
    • Incorrectly Identifying the Transverse Axis: For a hyperbola, the transverse axis corresponds to the positive term in the standard form equation, not necessarily the term with the larger denominator.
    • Errors in Asymptote Equations: Mixing up the numerator and denominator in the slope of the asymptotes. The slope is always a variation of \(\pm \frac{\text{rise}}{\text{run}}\) or \(\pm \frac{v_y}{v_x}\).
    • Errors in Completing the Square: When converting from general to standard form, failing to correctly account for factored coefficients when balancing the equation. For example, when changing \(9x^2 - 36x\) to \(9(x^2 - 4x + 4)\), you have added \(9 \times 4 = 36\) to that side of the equation.
    • Sign Errors with Center Coordinates: Incorrectly identifying the center \((h,k)\) from the standard form. For example, in \(\frac{(x-2)^2}{36}\), \(h=2\), and in \(\frac{(y+5)^2}{81}\), \(k=-5\).

    This page titled 15.2.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:48:09 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.