Hyperbolas

Front

What is a hyperbola?

Back

A hyperbola is a type of conic section that appears as two separate curves, known as branches, which are mirror images of each other. It is defined as the set of all points where the difference of the distances to two fixed points (foci) is constant.

Front

What are the vertices of a hyperbola?

Back

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. For a hyperbola centered at (h, k) with a horizontal transverse axis, the vertices are located at (h ± a, k), where 'a' is the distance from the center to each vertex.

Front

What are the asymptotes of a hyperbola?

Back

The asymptotes of a hyperbola are straight lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k) with a horizontal transverse axis, the equations of the asymptotes are y = k ± (b/a)(x - h), where 'a' and 'b' are the distances from the center to the vertices and co-vertices, respectively.

Front

How do you determine if a hyperbola is vertical or horizontal?

Back

A hyperbola is horizontal if its transverse axis is horizontal (opens left and right) and is represented by the equation (x - h)²/a² - (y - k)²/b² = 1. It is vertical if its transverse axis is vertical (opens up and down) and is represented by the equation (y - k)²/a² - (x - h)²/b² = 1.

Front

What is the standard form of a hyperbola with a horizontal transverse axis?

Back

The standard form of a hyperbola with a horizontal transverse axis is (x - h)²/a² - (y - k)²/b² = 1, where (h, k) is the center, 'a' is the distance to the vertices, and 'b' is the distance to the co-vertices.

Front

What is the standard form of a hyperbola with a vertical transverse axis?

Back

The standard form of a hyperbola with a vertical transverse axis is (y - k)²/a² - (x - h)²/b² = 1, where (h, k) is the center, 'a' is the distance to the vertices, and 'b' is the distance to the co-vertices.

Front

What is the relationship between the distances 'a', 'b', and 'c' in a hyperbola?

Back

In a hyperbola, the relationship is given by the equation c² = a² + b², where 'c' is the distance from the center to each focus, 'a' is the distance from the center to each vertex, and 'b' is the distance from the center to each co-vertex.

Front

How do you find the foci of a hyperbola?

Back

The foci of a hyperbola can be found using the formula (h ± c, k) for a horizontal hyperbola and (h, k ± c) for a vertical hyperbola, where 'c' is calculated using c² = a² + b².

Front

What is the transverse axis of a hyperbola?

Back

The transverse axis of a hyperbola is the line segment that connects the two vertices of the hyperbola. It is the axis along which the hyperbola opens.

Front

What is the conjugate axis of a hyperbola?

Back

The conjugate axis of a hyperbola is the line segment that connects the two co-vertices. It is perpendicular to the transverse axis and does not intersect the hyperbola.