Radius and Centre of Equation of Circle in Various Forms

Understanding the equation of a circle is crucial for solving many geometric problems. The equation of a circle can be represented in various forms, each with its own way of defining the radius and center of the circle. Below, we will explore the different forms of the equation of a circle and how to determine the radius and center from each.

Standard Form

The standard form of the equation of a circle is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

where $(h, k)$ is the center of the circle and $r$ is the radius.

Example:

Given the equation $(x - 3)^2 + (y + 2)^2 = 16$, the center is $(3, -2)$ and the radius is $\sqrt{16} = 4$.

General Form

The general form of the equation of a circle is:

$$x^2 + y^2 + Dx + Ey + F = 0$$

To find the center and radius from the general form, we complete the square for both $x$ and $y$.

Example:

Given the equation $x^2 + y^2 - 6x + 8y + 9 = 0$, we can rewrite it as:

$$(x^2 - 6x) + (y^2 + 8y) = -9$$

Complete the square:

$$(x - 3)^2 + (y + 4)^2 = 16$$

Now, it is in standard form, and we can see that the center is $(3, -4)$ and the radius is $\sqrt{16} = 4$.

Parametric Form

In the parametric form, the equation of a circle is expressed using a parameter, typically $t$, which represents the angle:

$$x = h + r\cos(t)$$ $$y = k + r\sin(t)$$

where $(h, k)$ is the center and $r$ is the radius.

Example:

For a circle with center at $(2, -1)$ and radius $5$, the parametric equations are:

$$x = 2 + 5\cos(t)$$ $$y = -1 + 5\sin(t)$$

Diameter Form

If the endpoints of a diameter of the circle are given by $(x_1, y_1)$ and $(x_2, y_2)$, then the equation of the circle can be written as:

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$

The center is the midpoint of the diameter, and the radius is half the distance between the endpoints.

Example:

Given the endpoints of a diameter at $(1, 2)$ and $(5, 6)$, the center is the midpoint, which is $\left(\frac{1+5}{2}, \frac{2+6}{2}\right) = (3, 4)$, and the radius is half the distance between the points, which can be calculated using the distance formula.

Differences and Important Points

Form	Equation	Center	Radius	Notes
Standard	$(x - h)^2 + (y - k)^2 = r^2$	$(h, k)$	$r$	Directly gives the center and radius.
General	$x^2 + y^2 + Dx + Ey + F = 0$	$(-D/2, -E/2)$	$\sqrt{r^2 - F}$	Requires completing the square to find center and radius.
Parametric	$x = h + r\cos(t)$, $y = k + r\sin(t)$	$(h, k)$	$r$	$t$ is a parameter representing the angle.
Diameter	$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$	Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$	Half the distance between $(x_1, y_1)$ and $(x_2, y_2)$	Derived from the endpoints of a diameter.

Each form has its own advantages depending on the given information and the context of the problem. Understanding how to convert between these forms and how to extract the center and radius is essential for solving circle-related problems in geometry and trigonometry.