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 the form x² + y² + Dx + Ey + F = 0, rearrange it to (x-h)² + (y-k)² = r², where h = -D/2 and k = -E/2.

Front

What is the radius of a circle given its equation in standard form?

Back

The radius r can be found from the equation (x-h)² + (y-k)² = r² by taking the square root of r².

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

Back

Completing the square involves rewriting a quadratic expression in the form (x-h)² to make it easier to identify the center and radius of the circle.

Front

What is the center of the circle given by the equation x² + y² - 6x + 4y - 12 = 0?

Back

The center is (3, -2).

Front

What is the radius of the circle given by the equation (x-1)² + (y+2)² = 16?

Back

The radius is 4, since r² = 16.

Front

How can you determine if a point lies inside, outside, or on 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.

Front

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

Back

The equation is x² + y² = 25.

Front

What is the significance of the coefficients in the general form of a circle's equation?

Back

The coefficients help determine the position and size of the circle, specifically the center and radius when rearranged.