Equations of Circles

Front

What is the standard form of a circle's equation?

Back

The standard form of a circle's equation 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 in general form?

Back

To find the center from the general form Ax² + Ay² + Bx + Cy + D = 0, rearrange to standard form and identify (h, k) as (-B/2A, -C/2A).

Front

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

Back

The radius is the square root of the value on the right side of the equation in standard form (x - h)² + (y - k)² = r².

Front

Convert the general equation x² + y² + 6x - 8y - 12 = 0 to standard form.

Back

(x + 3)² + (y - 4)² = 25.

Front

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

Back

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial, which helps in rewriting the equation in standard form.

Front

Given the equation x² + y² - 4x + 6y - 12 = 0, find the center and radius.

Back

Center: (2, -3), Radius: 5.

Front

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

Back

The coefficients help determine the center and radius of the circle when rearranged into standard form.

Front

How do 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, it's inside; if equal, it's on; if greater, it's outside.

Front

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

Back

The diameter is twice the radius (d = 2r).

Front

How can you identify the radius from the general form of a circle's equation?

Back

Rearrange the equation to standard form and calculate the radius as the square root of the constant term on the right side.