The standard form of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center and r is the radius. This form directly shows the distance from any point (x, y) to the center.

In the equation (x - 3)^2 + (y + 2)^2 = 16, the right side represents r². Since 16 is the square of 4, the radius r is 4.

The variables h and k in the standard equation represent the x and y coordinates of the center, respectively. This form makes it clear where the center of the circle is located.

For a circle centered at (-2, 5), the equation is written as (x + 2)^2 + (y - 5)^2 = r², where r² = 9 since the radius is 3. This correctly represents the translation of the circle's center.

In the standard circle equation, the constant on the right side is r². Here, 36 is the square of the radius, meaning the actual radius is √36, which equals 6.

By grouping x and y terms and completing the square, x² - 6x becomes (x - 3)² with an extra 9, and y² + 4y becomes (y + 2)² with an extra 4. Adjusting the equation yields (x - 3)² + (y + 2)² = 4.

Divide the entire equation by 9 to simplify and then complete the square for both x and y terms. This process reveals the center at (1, -2) and that r² equals 4, making the radius 2.

To rewrite the general equation in standard form, group the x and y terms and complete the square for each. This process involves adding the square of half the coefficient of x and y to both sides of the equation.

The radius of the circle is the distance between the center (2, -1) and the point (2, 3), which computes to 4. Substituting the center and radius into the standard equation gives (x - 2)² + (y + 1)² = 16.

The radius of a circle is the distance from its center to any point on its circumference. This is calculated using the distance formula: √[(x - h)² + (y - k)²].

Using the distance formula between the center (3, -k) and the point (-1, 2) and setting it equal to 5, two solutions for k are obtained. Considering the positive value yields k = 1.

Completing the square for the general equation reveals that the center of the circle is located at (-g, -f). This is a standard result derived from rewriting the equation in center-radius form.

To determine if a point lies on a circle, substitute the point into the circle's equation. If the left-hand side equals r², then the point is on the circumference.

Substitute the point (4, 6) into the equation to calculate k: (4 - 1)² + (6 - 2)² equals 9 + 16, which sums to 25. Hence, k is 25.

The current center is (3, -4), so subtracting 3 from the x-coordinate and adding 4 to the y-coordinate (i.e., translating by (-3, 4)) will relocate the center to (0, 0). The radius remains unchanged under translation.

A circle tangent to the x-axis has its radius equal to the y-coordinate of its center. By setting the center on the line y = 2x - 3 and using the distance from the center to the given point (4, 5), one can solve for the x-coordinate. Choosing the solution that yields a positive radius leads to the equation provided.

The distance from the center (a, b) to the origin is √(a² + b²), which is exactly the radius as indicated by the equation. Therefore, the circle will always pass through the origin.

The center of the circle is the midpoint of the diameter, which calculates to (1, 1). The radius is half the distance between the endpoints; with the total distance being 10, the radius is 5, yielding the equation (x - 1)² + (y - 1)² = 25.

Let the center be (2, k) with the circle tangent to the line y = x + 3, which implies the distance from (2, k) to the line equals the radius. Using the distance formula for the point (5, 9) and equating it to the tangent distance leads to a quadratic in k; selecting k = 13 - √14 provides the consistent solution and corresponding radius.

The distance from the circle's center (4, -m) to the line y = 2x - 5 must equal the radius, 7. Writing the line in standard form and applying the distance formula gives |m + 3|/√5 = 7. Since m > -3, solving yields m = 7√5 - 3.