Circles and Ellipses

Front

What are the major vertices of an ellipse?

Back

The major vertices of an ellipse are the points where the ellipse intersects its major axis. For example, in the ellipse with the equation (x^2/16) + (y^2/9) = 1, the major vertices are (4, 0) and (-4, 0).

Front

What is the standard equation of a circle?

Back

The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

Front

How do you find the radius of a circle from its equation?

Back

To find the radius of a circle from its equation in standard form (x - h)² + (y - k)² = r², take the square root of r².

Front

What are the coordinates of the center of a circle given by the equation (x - 4)² + (y - 3)² = 25?

Back

The coordinates of the center are (4, 3).

Front

What is the relationship between the major and minor axes of an ellipse?

Back

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest diameter. The lengths of the axes are determined by the values under the squared terms in the standard form of the ellipse equation.

Front

How do you graph a circle given its equation?

Back

To graph a circle, identify the center (h, k) from the equation (x - h)² + (y - k)² = r², then plot points at a distance r from the center in all directions.

Front

What is the formula for the area of a circle?

Back

The area A of a circle is given by the formula A = πr², where r is the radius.

Front

What is the formula for the area of an ellipse?

Back

The area A of an ellipse is given by the formula A = πab, where a is the semi-major axis and b is the semi-minor axis.

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 major and minor axes are equal.

Front

How do you determine the foci of an ellipse?

Back

The foci of an ellipse can be found using the formula c = √(a² - b²), where a is the semi-major axis and b is the semi-minor axis. The foci are located at (±c, 0) for horizontal ellipses and (0, ±c) for vertical ellipses.