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Ellipses: Standard Form quiz

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  • What is the standard equation for a horizontal ellipse centered at the origin?
    The equation is (x²/a²) + (y²/b²) = 1, where a > b.
  • How do you determine if an ellipse is horizontal or vertical from its equation?
    If a² is under x², it's horizontal; if a² is under y², it's vertical.
  • What do the variables a and b represent in the ellipse equation?
    a is the semi-major axis (longest distance from center), and b is the semi-minor axis (shortest distance from center).
  • How do you find the vertices of a horizontal ellipse centered at the origin?
    The vertices are at (a, 0) and (-a, 0).
  • What is the formula to find the distance from the center to the foci of an ellipse?
    Use c² = a² - b², where c is the distance from the center to each focus.
  • Where are the foci located for a vertical ellipse centered at the origin?
    The foci are at (0, c) and (0, -c), where c = sqrt(a² - b²).
  • How does the equation of an ellipse change if it is centered at (h, k) instead of the origin?
    The equation becomes ((x-h)²/a²) + ((y-k)²/b²) = 1.
  • How do you identify the center of an ellipse from its equation in standard form?
    The center is at (h, k), where h and k are the values subtracted from x and y in the equation.
  • What is the relationship between the circle equation and the ellipse equation?
    The circle equation is a special case of the ellipse where a = b (radius), giving symmetry in all directions.
  • How do you find the coordinates of the vertices for a vertical ellipse centered at (h, k)?
    The vertices are at (h, k+a) and (h, k-a).
  • How do you find the coordinates of the foci for a horizontal ellipse centered at (h, k)?
    The foci are at (h+c, k) and (h-c, k), where c = sqrt(a² - b²).
  • What does the sum of the distances from any point on the ellipse to the two foci equal?
    It equals 2a, the length of the major axis.
  • How do you determine which denominator is a² and which is b² in the ellipse equation?
    a² is always the larger denominator, and b² is the smaller one.
  • What are the coordinates of the 'b points' (co-vertices) for a vertical ellipse centered at (h, k)?
    They are at (h+b, k) and (h-b, k).
  • How do you graph an ellipse given its standard form equation?
    Identify the center (h, k), determine a and b, plot the vertices and co-vertices, and connect them with a smooth curve.