Ellipses

Front

What is the standard equation of a circle with center (h, k) and radius r?

Back

The standard equation of a circle is (x - h)² + (y - k)² = r².

Front

How do you identify the center and radius of a circle from its equation?

Back

The center (h, k) can be found from the equation (x - h)² + (y - k)² = r², and the radius r is the square root of the constant on the right side.

Front

What is the general form of the equation of an ellipse?

Back

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

Front

How do you determine the lengths of the semi-major and semi-minor axes from the equation of an ellipse?

Back

In the equation (x - h)²/a² + (y - k)²/b² = 1, a is the length of the semi-major axis and b is the length of the semi-minor axis.

Front

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

Back

In an ellipse, the length of the semi-major axis (a) is always greater than or equal to the length of the semi-minor axis (b).

Front

What is the definition of a conic section?

Back

A conic section is a curve obtained by intersecting a cone with a plane. The types of conic sections include circles, ellipses, parabolas, and hyperbolas.

Front

What is the difference between a circle and an ellipse?

Back

A circle is a special case of an ellipse where the lengths of the semi-major and semi-minor axes are equal (a = b).

Front

How do you factor the expression x² - a²?

Back

The expression x² - a² can be factored using the difference of squares formula: x² - a² = (x - a)(x + a).

Front

What is the significance of the center of an ellipse?

Back

The center of an ellipse is the midpoint between its two foci and is the point from which the ellipse is symmetrically defined.

Front

What are the coordinates of the foci of an ellipse?

Back

The coordinates of the foci of an ellipse can be found using the formula (h ± c, k) for horizontal ellipses and (h, k ± c) for vertical ellipses, where c = √(a² - b²).