Conic Sections Review (Circles, Ellipses, Hyperbolas)

Front

What is the standard form of the equation of a circle?

Back

(x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Front

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

Back

The radius is the square root of the constant on the right side of the equation in standard form.

Front

What are the characteristics of an ellipse?

Back

An ellipse has two foci, a major axis, a minor axis, and is defined by the sum of distances from any point on the ellipse to the two foci being constant.

Front

What is the difference 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.

Front

How do you determine the orientation of an ellipse?

Back

If the major axis is horizontal, the equation is of the form (x²/a²) + (y²/b²) = 1. If vertical, it is (y²/a²) + (x²/b²) = 1.

Front

What is the standard form of the equation of a hyperbola?

Back

(x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1, where (h, k) is the center.

Front

What are the vertices of a hyperbola?

Back

The vertices are the points where the hyperbola intersects its transverse axis.

Front

How do you find the center of a hyperbola given its vertices?

Back

The center is the midpoint of the line segment connecting the vertices.

Front

What is the relationship between the foci and vertices of a hyperbola?

Back

The distance from the center to each focus is greater than the distance from the center to each vertex.

Front

What is the equation of a circle with center (2, -4) and radius 3?

Back

(x - 2)² + (y + 4)² = 9.