Equations of Circles

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. The center is (h, k).

Front

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

Back

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

Front

What does the equation (x + 1)² + (y - 5)² = 25 represent?

Back

This equation represents a circle with center (-1, 5) and radius 5.

Front

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

Back

To determine the position of a point (x₀, y₀) relative to a circle with center (h, k) and radius r, calculate the distance from the point to the center: d = √((x₀ - h)² + (y₀ - k)²). If d < r, the point is inside; if d = r, it is on the circle; if d > r, it is outside.

Front

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

Back

The general form of the equation of a circle is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants.

Front

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

Back

To convert from general form x² + y² + Dx + Ey + F = 0 to standard form, complete the square for the x and y terms.

Front

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

Back

The radius determines the size of the circle; it is the distance from the center to any point on the circle.

Front

If a circle has a center at (0, 0) and a radius of 1, what is its equation?

Back

The equation is x² + y² = 1.

Front

What is the relationship between the diameter and radius of a circle?

Back

The diameter is twice the radius; d = 2r.