General Equation of a Circle

Front

What is the general equation of a circle?

Back

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

Front

How do you convert the general form of a circle to center-radius form?

Back

To convert from general form to center-radius form, you must complete the square.

Front

What is the center-radius form of a circle?

Back

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

Front

What does completing the square involve?

Back

Completing the square involves rearranging a quadratic equation to form a perfect square trinomial.

Front

What is the center of a circle in the equation x² + y² + 6x - 12y + 20 = 0?

Back

The center of the circle is (-3, 6).

Front

What is the radius of a circle in the equation (x + 3)² + (y - 6)² = 25?

Back

The radius of the circle is 5, since r² = 25.

Front

How do you find the radius from the general form of a circle?

Back

To find the radius from the general form, rearrange the equation to center-radius form and solve for r.

Front

What is the significance of the coefficients D, E, and F in the general equation of a circle?

Back

The coefficients D and E relate to the center of the circle, while F helps determine the radius.

Front

What is the first step in converting the general form of a circle to standard form?

Back

The first step is to group the x and y terms together.

Front

What is the standard form of the circle given by the equation x² + y² + 6x - 12y + 20 = 0?

Back

The standard form is (x + 3)² + (y - 6)² = 25.