Conics: Ellipses

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 identify the center of an ellipse from its equation?

Back

The center (h, k) can be identified from the standard form (x-h)²/a² + (y-k)²/b² = 1.

Front

What do the variables a and b represent in the context of an ellipse?

Back

In an ellipse, 'a' represents the length of the semi-major axis, and 'b' represents the length of the semi-minor axis.

Front

How do you find the foci of an ellipse?

Back

The foci can be found using the formula c = √(a² - b²), where c is the distance from the center to each focus.

Front

What is the relationship between the lengths of the axes and the foci in an ellipse?

Back

The distance to the foci (c) is always less than the length of the semi-major axis (a), i.e., c < a.

Front

What are the vertices of an ellipse?

Back

The vertices are the endpoints of the major axis, located at (h ± a, k) for horizontal ellipses and (h, k ± a) for vertical ellipses.

Front

How do you determine the orientation of an ellipse from its equation?

Back

If a² > b², the ellipse is horizontal; if b² > a², the ellipse is vertical.

Front

What is the formula for the length of the major axis of an ellipse?

Back

The length of the major axis is 2a.

Front

What is the formula for the length of the minor axis of an ellipse?

Back

The length of the minor axis is 2b.

Front

How do you write the equation of an ellipse given its center and lengths of axes?

Back

Use the standard form: (x-h)²/a² + (y-k)²/b² = 1, substituting h, k, a, and b.