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.

Front

What is the radius of a circle given the equation (x - 3)² + (y + 4)² = 16?

Back

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

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 if the radius is 5?

Back

The diameter is 10, since diameter = 2 * radius.

Front

Given the equation (x + 2)² + (y - 1)² = 25, what is the center and radius?

Back

Center: (-2, 1), Radius: 5.

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

If a circle has the equation (x - 1)² + (y + 3)² = 9, what is the radius?

Back

The radius is 3, since r² = 9, thus r = √9 = 3.

Front

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

Back

The diameter is twice the radius: Diameter = 2 * Radius.

Front

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

Back

Substitute the point's coordinates into the circle's equation. If the result is less than r², the point is inside; if equal, on; if greater, outside.