Ellipse and Hyperbolas

Front

What is the standard equation of an ellipse?

Back

The standard equation of an ellipse centered at (h, k) is: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) where a is the semi-major axis and b is the semi-minor axis.

Front

What is the standard form of a hyperbola?

Back

The standard equation of a hyperbola centered at (h, k) is: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) for a horizontal hyperbola, and \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \) for a vertical hyperbola.

Front

What is the center of an ellipse?

Back

The center of an ellipse is the point (h, k) in the standard equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

Front

How do you find the foci of an ellipse?

Back

The foci of an ellipse are located at (h ± c, k) for horizontal ellipses and (h, k ± c) for vertical ellipses, where \( c = \sqrt{a^2 - b^2} \).

Front

What is the difference between an ellipse and a hyperbola?

Back

An ellipse is a set of points where the sum of the distances to two foci is constant, while a hyperbola is a set of points where the difference of the distances to two foci is constant.

Front

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

Back

The equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Front

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

Back

In an ellipse, the relationship is given by \( c^2 = a^2 - b^2 \), where c is the distance from the center to each focus.

Front

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

Back

In a hyperbola, the relationship is given by \( c^2 = a^2 + b^2 \), where c is the distance from the center to each focus.

Front

How do you graph an ellipse?

Back

To graph an ellipse, identify the center (h, k), determine the lengths of the semi-major (a) and semi-minor (b) axes, and plot the vertices and co-vertices.

Front

What is the directrix of a hyperbola?

Back

The directrix of a hyperbola is a line used to define the hyperbola, located at a distance of \( \frac{a^2}{c} \) from the center.