Ellipse and Hyperbolas

Front

What is the standard form of the equation of a hyperbola centered at (h, k)?

Back

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

Front

What is the definition of an ellipse?

Back

An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (foci) is constant.

Front

What is the standard form of the equation of an ellipse centered at (h, k)?

Back

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

Front

How do you find the foci of an ellipse?

Back

The foci are located at (h±c, k) for horizontal ellipses and (h, k±c) for vertical ellipses, where c = √(a² - b²).

Front

What is the relationship between a, b, and c in an ellipse?

Back

In an ellipse, c² = a² - b², where a is the semi-major axis and b is the semi-minor axis.

Front

What is the definition of a hyperbola?

Back

A hyperbola is a set of points in a plane where the absolute difference of the distances from two fixed points (foci) is constant.

Front

How do you find the asymptotes of a hyperbola?

Back

The asymptotes of a hyperbola centered at (h, k) are given by the equations y - k = ±(b/a)(x - h) for a horizontal hyperbola and y - k = ±(a/b)(x - h) for a vertical hyperbola.

Front

What is the difference between a major axis and a minor axis in an ellipse?

Back

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest diameter.

Front

What is the equation of a hyperbola with vertices at (±a, 0) and foci at (±c, 0)?

Back

The equation is x²/a² - y²/b² = 1, where c² = a² + b².

Front

How do you convert the general equation of an ellipse to standard form?

Back

To convert, complete the square for both x and y terms and rearrange to match the standard form.