Final answer:
To convert a circle's equation to standard form and find its center and radius, complete the square for both x and y terms, resulting in an equation of the form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Step-by-step explanation:
To rewrite a circle's equation in standard form and find its center and radius, start by ensuring the equation is in the form (x - h)² + (y - k)² = r².
The (h, k) represents the center of the circle, and r is the radius. To convert an equation to this form, you may need to complete the square for both the x and y-terms. Once the equation is in standard form, the center can be read directly as the coordinates opposite in sign to those inside the parenthesis. The radius is the square root of the number on the right side of the equation.
For example, for an equation x² + y² - 6x - 8y + 9 = 0, you would complete the square for x and y separately, and adjust the right side of the equation accordingly to find the standard form.
The center is at the point (h, k), and you can use r to calculate the radius. When the circle is centered on the origin, the equation simplifies since h and k would both be zero.