Circles in Standard Form

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 given its equation in standard form?

Back

The center of a circle in standard form (x - h)² + (y - k)² = r² is given by the point (h, k).

Front

What is the radius of a circle if its equation is (x - 3)² + (y + 2)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 distance between the center and a point on the circle?

Back

The distance from the center (h, k) to a point (x, y) on the circle is equal to the radius r.

Front

If the endpoints of a diameter are given, how do you find the center of the circle?

Back

The center of the circle is the midpoint of the diameter, calculated as ((x1 + x2)/2, (y1 + y2)/2).

Front

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

Back

The equation is x² + y² = 25.

Front

How do you determine the radius from the center and a point on the circle?

Back

The radius can be found using the distance formula: r = √((x - h)² + (y - k)²), where (h, k) is the center.

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

How can you identify the center and radius from the equation x² + y² + 6x - 4y = 12?

Back

First, rewrite in standard form by completing the square: center = (-3, 2), radius = 5.