Hyperbolas

Front

What is a hyperbola?

Back

A hyperbola is a type of conic section that appears as two separate curves, defined as the set of all points where the difference of the distances to two fixed points (foci) is constant.

Front

What is the standard form of a hyperbola?

Back

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

Front

What are the asymptotes of a hyperbola?

Back

The asymptotes of a hyperbola are lines that the hyperbola approaches but never touches. For the standard form (x-h)²/a² - (y-k)²/b² = 1, the asymptotes are given by y = k ± (b/a)(x-h).

Front

How do you find the vertices of a hyperbola?

Back

The vertices of a hyperbola are located at (h±a, k) for a horizontal hyperbola and (h, k±a) for a vertical hyperbola.

Front

What is the relationship between the foci and vertices of a hyperbola?

Back

The distance from the center to each focus (c) is related to the distance to each vertex (a) by the equation c² = a² + b².

Front

How do you convert the general equation of a hyperbola to standard form?

Back

To convert to standard form, rearrange the equation to isolate the hyperbola terms, complete the square for x and y terms, and then divide by the constant to set the equation equal to 1.

Front

What is the equation of the asymptotes for the hyperbola 4y² - 16x² = 64?

Back

The asymptotes are y = ±2x.

Front

Identify the vertices of the hyperbola defined by the equation (x-10)²/16 - (y+8)²/4 = 1.

Back

The vertices are (10±4, -8) or (14, -8) and (6, -8).

Front

What is the significance of the transverse axis in a hyperbola?

Back

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

Front

What is the conjugate axis of a hyperbola?

Back

The conjugate axis is the line segment perpendicular to the transverse axis at the center of the hyperbola, with length 2b.