Circle Equations

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

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

Back

The center of a circle in the equation (x - h)² + (y - k)² = r² is the point (h, k) where the circle is centered.

Front

How do you determine the radius from the circle's equation?

Back

In the equation (x - h)² + (y - k)² = r², the radius r is the square root of the value on the right side of the equation.

Front

What is the center of the circle given by the equation (x - 3)² + (y + 4)² = 16?

Back

The center of the circle is (3, -4).

Front

How can you tell if a point lies inside, outside, or on a circle?

Back

To determine the position of a point (x₀, y₀) relative to a circle with equation (x - h)² + (y - k)² = r², calculate (x₀ - h)² + (y₀ - k)². If the result is less than r², the point is inside; if equal, on the circle; if greater, outside.

Front

What is the radius of the circle defined by the equation x² + y² = 25?

Back

The radius of the circle is 5, since r² = 25.

Front

Write the equation of a circle with center (0, 0) and radius 4.

Back

The equation is x² + y² = 16.

Front

What is the equation of a circle with center (2, -3) and radius 5?

Back

The equation is (x - 2)² + (y + 3)² = 25.

Front

What does the term 'conic section' refer to?

Back

A conic section is a curve obtained by intersecting a cone with a plane, which includes circles, ellipses, parabolas, and hyperbolas.

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.