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Ellipses: Standard Form quiz #1

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  • Given the graph of an ellipse, how can you determine which equation represents this ellipse in standard form?
    To determine the equation of an ellipse from its graph, identify the center (h, k), the lengths of the semi-major axis (a) and semi-minor axis (b), and the orientation (horizontal or vertical). The standard form is (x−h)²/a² + (y−k)²/b² = 1 for a horizontal ellipse (a > b, a under x), or (x−h)²/b² + (y−k)²/a² = 1 for a vertical ellipse (a > b, a under y).
  • How do you write the standard form equation for an ellipse based on its graph?
    First, find the center (h, k) of the ellipse. Next, measure the lengths of the semi-major axis (a) and semi-minor axis (b). If the major axis is horizontal, use (x−h)²/a² + (y−k)²/b² = 1. If the major axis is vertical, use (x−h)²/b² + (y−k)²/a² = 1.
  • What geometric process creates an ellipse from a cone?
    An ellipse is formed by slicing a three-dimensional cone with a two-dimensional plane at a slight angle. This produces a shape defined by two axes: the semi-major and semi-minor axes.
  • How do the distances from the center to points on an ellipse differ from those on a circle?
    On a circle, the distance from the center to any point is always the same, called the radius. On an ellipse, the distances vary and are defined by the semi-major axis (a) and semi-minor axis (b).
  • What is the relationship between a, b, and c in an ellipse?
    The relationship is given by c² = a² - b², where c is the distance from the center to each focus. This equation helps locate the foci of the ellipse.
  • Where are the vertices and foci located for a horizontal ellipse centered at the origin?
    For a horizontal ellipse at the origin, the vertices are at (a, 0) and (−a, 0), and the foci are at (c, 0) and (−c, 0). Both sets of points lie along the x-axis.
  • How do you determine the orientation of an ellipse from its equation?
    The orientation is determined by which variable has the larger denominator; if a² is under x, the ellipse is horizontal, and if a² is under y, it is vertical. The major axis aligns with the variable having the larger denominator.
  • What does the center (h, k) represent in the equation of an ellipse not at the origin?
    The center (h, k) indicates the horizontal and vertical shift of the ellipse from the origin. It is found by identifying the values subtracted from x and y in the equation.
  • How do you find the coordinates of the vertices for a vertical ellipse centered at (h, k)?
    For a vertical ellipse, the vertices are at (h, k + a) and (h, k − a), where a is the length of the semi-major axis. These points are found by adding and subtracting a from the k value.
  • What is the constant property involving the foci and any point on an ellipse?
    For any point on an ellipse, the sum of the distances from that point to each focus is always constant. This constant equals the length of the major axis, 2a.
  • What is the eccentricity of an ellipse that is completely flat (i.e., has collapsed into a line segment)?
    The eccentricity of a completely flat ellipse, where the semi-minor axis b is zero, is 1. Eccentricity (e) is defined as c/a, where c = sqrt(a^2 - b^2). If b = 0, then c = a, so e = a/a = 1.
  • How are the directrices of an ellipse defined in relation to its axes and foci?
    The directrices of an ellipse are lines perpendicular to the major axis, located at a distance of a/e from the center, where a is the semi-major axis and e is the eccentricity. Each directrix is associated with one of the foci.