Equations of Circles Review

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 in the equation (x+1)² + (y-2)² = 9?

Back

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

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 does the equation (x-4)² + (y+4)² = 25 represent?

Back

This represents a circle with center (4, -4) and radius 5.

Front

What is the significance of the number on the right side of the equation in standard form?

Back

The number on the right side of the equation (r²) represents the square of the radius of the circle.

Front

How can 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, it's inside; equal to r², it's on; greater than r², it's outside.

Front

What is the center of the circle in the equation (x-3)² + (y-2)² = 16?

Back

The center of the circle is (3, 2).

Front

What is the radius of the circle represented by the equation (x-5)² + (y+1)² = 36?

Back

The radius is 6, since r² = 36, thus r = √36 = 6.

Front

How do you identify the center and radius from the equation x² + y² - 4x + 6y - 12 = 0?

Back

First, rearrange to standard form by completing the square: (x-2)² + (y+3)² = 25. Center is (2, -3) and radius is 5.