Ellipses

Front

What is the definition of an ellipse?

Back

An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (foci) is constant.

Front

What are the standard forms of the equation of an ellipse?

Back

The standard forms are: (x-h)²/a² + (y-k)²/b² = 1 (horizontal ellipse) and (x-h)²/b² + (y-k)²/a² = 1 (vertical ellipse), where (h,k) is the center.

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, a is the semi-major axis, and b is the semi-minor axis.

Front

What is the relationship between the axes of an ellipse?

Back

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest diameter, and they are perpendicular to each other.

Front

What is the equation of a circle in standard form?

Back

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

Front

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

Back

The radius can be determined from the equation (x-h)² + (y-k)² = r² by taking the square root of r².

Front

What is the significance of the center in the equation of a circle?

Back

The center (h,k) is the point from which all points on the circle are equidistant, defining the circle's location in the coordinate plane.

Front

How do you convert the general equation of a circle to standard form?

Back

To convert, complete the square for the x and y terms in the general equation Ax² + Ay² + Bx + Cy + D = 0.

Front

What is the distance between the foci of an ellipse?

Back

The distance between the foci is 2c, where c is calculated using c = √(a² - b²).

Front

What is the relationship between the semi-major axis and semi-minor axis in an ellipse?

Back

In an ellipse, a ≥ b, where a is the length of the semi-major axis and b is the length of the semi-minor axis.