Conics: Parabolas-Ellipses-Hyperbolas

Front

What is a parabola?

Back

A parabola is a U-shaped curve that is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Front

What is the standard form of a parabola's equation?

Back

The standard form of a parabola that opens upwards or downwards is (x-h)² = 4p(y-k), where (h,k) is the vertex and p is the distance from the vertex to the focus.

Front

Define the vertex of a parabola.

Back

The vertex of a parabola is the point where the curve changes direction; it is the highest or lowest point of the parabola.

Front

What is the focus of a parabola?

Back

The focus of a parabola is a fixed point located inside the curve, which, along with the directrix, defines the parabola.

Front

What is the directrix of a parabola?

Back

The directrix of a parabola is a fixed line used in the definition of the parabola; every point on the parabola is equidistant from the focus and the directrix.

Front

What is an ellipse?

Back

An ellipse is a closed curve that results from the intersection of a cone with a plane that is inclined to the axis of the cone, where the sum of the distances from any point on the ellipse to two fixed points (the foci) is constant.

Front

What is the standard form of an ellipse's equation?

Back

The standard form of an ellipse centered at (h,k) is (x-h)²/a² + (y-k)²/b² = 1, where 2a is the length of the major axis and 2b is the length of the minor axis.

Front

Define the foci of an ellipse.

Back

The foci of an ellipse are two fixed points located along the major axis, such that the sum of the distances from any point on the ellipse to the foci is constant.

Front

What is a hyperbola?

Back

A hyperbola is a type of conic section that is formed by the intersection of a plane with both halves of a double cone, characterized by two separate curves called branches.

Front

What is the standard form of a hyperbola's equation?

Back

The standard form of a hyperbola centered at (h,k) is (x-h)²/a² - (y-k)²/b² = 1 for a horizontal hyperbola, and (y-k)²/a² - (x-h)²/b² = 1 for a vertical hyperbola.