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Conic Sections quiz

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  • What is a conic section and how are the four main types formed?
    A conic section is a shape formed by slicing a three-dimensional cone with a two-dimensional plane. The four main types—circle, ellipse, parabola, and hyperbola—are formed by slicing the cone at different angles.
  • What is the standard equation of a circle centered at the origin?
    The standard equation is x² + y² = r², where r is the radius of the circle.
  • How do you find the center and radius of a circle from its equation in standard form?
    The center is at (h, k) from the terms (x-h)² and (y-k)², and the radius is the square root of the constant on the right side.
  • What is the process called for converting a circle's general equation to standard form?
    The process is called completing the square, and it is done separately for x and y terms.
  • How is the equation of an ellipse at the origin written, and what do a and b represent?
    The equation is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis.
  • How do you determine if an ellipse is horizontal or vertical?
    If a² is under x², the ellipse is horizontal; if a² is under y², the ellipse is vertical.
  • What is the relationship between a, b, and c for the foci of an ellipse?
    The relationship is c² = a² - b², where c is the distance from the center to each focus.
  • How does the equation of an ellipse change when its center is not at the origin?
    The equation becomes (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the new center.
  • What is the standard form equation for a vertical parabola with vertex at (h, k)?
    The equation is (x-h)² = 4p(y-k), where p is the distance from the vertex to the focus.
  • How do you find the focus and directrix of a parabola given its equation?
    The focus is p units from the vertex in the direction the parabola opens, and the directrix is p units in the opposite direction.
  • How does the equation of a horizontal parabola differ from a vertical one?
    For a horizontal parabola, the equation is (y-k)² = 4p(x-h); x and y are switched compared to the vertical case.
  • What is the main difference between the equations of an ellipse and a hyperbola?
    An ellipse uses a plus sign (x²/a² + y²/b² = 1), while a hyperbola uses a minus sign (x²/a² - y²/b² = 1 or vice versa).
  • How do you find the vertices and foci of a hyperbola centered at the origin?
    Vertices are a units from the center along the transverse axis; foci are c units from the center, where c² = a² + b².
  • What is the formula for the asymptotes of a hyperbola centered at the origin?
    For a horizontal hyperbola, the asymptotes are y = ±(b/a)x; for a vertical hyperbola, they are y = ±(a/b)x.
  • How does the equation of a hyperbola change when its center is at (h, k)?
    The equation becomes ((x-h)²/a²) - ((y-k)²/b²) = 1 for horizontal or ((y-k)²/a²) - ((x-h)²/b²) = 1 for vertical hyperbolas.