In Exercises 53-64, complete the square and write the equation in standard form. Then give the c

step 1 We need to write the equation of the circle in standard form. Find the center in the radius and

In Exercises 53-64, complete the square and write the...

Key Concepts

Graphing a Circle

Graphing a circle involves plotting its center on the coordinate plane and drawing a circle using the radius as the distance from the center. This process visualizes the set of all points that are a fixed distance (the radius) away from the center, making the geometric properties of the circle clear. Graphing aids in the interpretation of the circle’s position and size relative to other objects in the plane.

Center and Radius

The center and radius of a circle provide essential geometric information: the center represents the fixed point equidistant from all points on the circle, while the radius is the constant distance from the center to any point on the circle's circumference. Identifying these parameters is crucial for understanding and graphing the circle.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square trinomial plus or minus a constant. This method involves adding and subtracting the appropriate constant term so that an expression like x² + bx becomes (x - h)² for some value h. It is essential for rewriting equations of conic sections in their standard forms and for solving quadratic equations.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents the radius. This form makes it straightforward to identify key properties of the circle such as its center and radius, and it is derived by rearranging and completing the square on the general form of the equation.

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