In Exercises 13-20, write the equation of the circle in standard form. Then identify its center

step 1 This is 19 number question 4x square plus 4 y square plus 12 x minus 24y plus 41 equal to 0 okay

In Exercises 13-20, write the equation of the circle in...

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Key Concepts

Center

The center of a circle refers to the point from which every point on the circle is equidistant. In the standard equation (x - h)² + (y - k)² = r², the coordinates (h, k) specify the center. This concept is fundamental in coordinate geometry as it helps in analysis and translation of the circle in the plane.

Radius

The radius of a circle is the constant distance from the center to any point on the circle. In the equation (x - h)² + (y - k)² = r², the value r, once obtained by taking the square root of the constant on the right side of the equation, represents this distance. Understanding the radius is crucial for determining the size of the circle and for further geometric computations.

Standard Form of a Circle

The standard form of a circle's equation is expressed as (x - h)² + (y - k)² = r². In this form, (h, k) represents the center of the circle, and r is the radius. This representation makes it straightforward to identify the circle's key geometric properties directly from the equation.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial. In the context of circle equations, this process is essential for rewriting a general second-degree equation into the standard form, thereby revealing the center and radius of the circle.

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