Circles and ellipses

Front

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

Back

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

Front

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

Back

To find the center of a circle from its equation in standard form (x - h)² + (y - k)² = r², identify the values of h and k.

Front

What is the radius of a circle given the equation (x - 2)² + (y + 3)² = 16?

Back

The radius is 4 units, since r² = 16, thus r = √16 = 4.

Front

What are the coordinates of the center of the circle given by the equation (x + 5)² + (y - 1)² = 9?

Back

The center is (-5, 1).

Front

What is the relationship between the radius and 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 equation of a circle with center (0, 0) and radius 3?

Back

The equation is x² + y² = 9.

Front

How do you determine if a point lies inside, on, or outside a circle?

Back

Calculate the distance from the point to the center. If the distance is less than the radius, the point is inside; if equal, on; if greater, outside.

Front

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

Back

The general form is x² + y² + Dx + Ey + F = 0.

Front

What is an ellipse?

Back

An ellipse is a set of points where the sum of the distances from two fixed points (foci) is constant.

Front

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

Back

The standard form 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.