Equations of Circles

Front

What is the standard equation of a circle?

Back

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

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 coordinates (h, k).

Front

What does the radius of a circle represent in its equation?

Back

The radius of a circle in the equation (x - h)² + (y - k)² = r² is the distance from the center (h, k) to any point on the circle, represented by r.

Front

If the equation of a circle is (x + 3)² + (y - 2)² = 25, what is the radius?

Back

The radius is 5 units, since r² = 25, thus r = √25 = 5.

Front

What is the center of the circle given by the equation (x - 1)² + (y + 4)² = 36?

Back

The center is (1, -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 relationship between the diameter and radius of a circle?

Back

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

Front

If a circle has a center at (0, 0) and a radius of 3, what is its equation?

Back

The equation is x² + y² = 9.

Front

What is the significance of the number on the right side of the circle's equation?

Back

The number on the right side of the equation (x - h)² + (y - k)² = r² represents the square of the radius (r²).

Front

How can 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, the point is inside; if equal, on the circle; if greater, outside.