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Ellipses quiz

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  • What is the definition of an ellipse in terms of distances to two points?
    An ellipse is the set of all points where the sum of the distances to two fixed points, called foci, is constant.
  • How does the definition of a circle differ from that of an ellipse?
    A circle is defined by all points at a constant distance (radius) from one center point, while an ellipse uses two foci and the sum of distances to them.
  • What are the foci of an ellipse?
    The foci are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to each focus is constant.
  • What is the standard equation of an ellipse centered at the origin?
    The standard equation is x²/a² + y²/b² = 1, where a and b are the horizontal and vertical distances from the center.
  • In the equation x²/a² + y²/b² = 1, what do 'a' and 'b' represent?
    'a' is the distance from the center to the ellipse along the x-axis, and 'b' is the distance along the y-axis.
  • How do you determine if an ellipse is stretched vertically or horizontally?
    If b > a, the ellipse is stretched vertically; if a > b, it is stretched horizontally.
  • What is the equation of a vertical ellipse with a = 3 and b = 4?
    The equation is x²/9 + y²/16 = 1.
  • What is the equation of a horizontal ellipse with a = 4 and b = 3?
    The equation is x²/16 + y²/9 = 1.
  • How do you graph an ellipse centered at the origin?
    Plot points a units left and right, and b units up and down from the center, then connect them smoothly.
  • What is the main difference between the equations of a circle and an ellipse?
    A circle has one squared distance (radius), while an ellipse has two different squared distances (a and b).
  • What is the sum of the distances from any point on an ellipse to the two foci?
    It is always the same constant value for every point on the ellipse.
  • If an ellipse is centered at the origin and a = b, what shape does it become?
    It becomes a circle.
  • What is the first step in sketching the graph of an ellipse?
    Find the values of a and b from the equation.
  • How do you find the values of a and b from the equation x²/a² + y²/b² = 1?
    Take the square root of the denominators to get a and b.
  • What is the orientation of the foci in a vertical ellipse?
    The foci are oriented along the y-axis in a vertical ellipse.