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

The center of a circle in the equation (x - h)² + (y - k)² = r² is given by the point (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, thus r = √16 = 4.

Front

If the center of a circle is at (0, 0) and the radius is 5, what is the equation of the circle?

Back

The equation is x² + y² = 25.

Front

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

Back

The diameter is twice the radius, so d = 2r.

Front

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

Back

To convert from general form Ax² + Ay² + Bx + Cy + D = 0 to standard form, complete the square for x and y.

Front

What is the diameter of a circle with the equation (x - 1)² + (y + 2)² = 36?

Back

The diameter is 12, since the radius is √36 = 6, and d = 2r.

Front

What does the term 'completing the square' mean in the context of circles?

Back

Completing the square is a method used to rewrite a quadratic equation in standard form by transforming it into a perfect square trinomial.

Front

What is the center of the circle represented by the equation (x - 4)² + (y + 1)² = 9?

Back

The center is (4, -1).

Front

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

Back

Substitute the point (x, y) into the circle's equation. If the result is less than r², it's inside; equal to r², it's on; greater than r², it's outside.