To find the center and radius of a sphere, you need its algebraic equation. The method depends on whether the equation is in its standard form or general form.
Finding the Center and Radius from Standard Form
The most straightforward way to determine a sphere's center and radius is when its equation is presented in the standard form.
The equation of a sphere in standard form is:
$$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$$
Here's what each part represents:
- The point $(\mathbf{h}, \mathbf{k}, \mathbf{l})$ represents the coordinates of the center of the sphere.
- The variable $\mathbf{r}$ represents the radius of the sphere.
It's crucial to note that the right side of the equation is $r^2$, so to find the actual radius $r$, you must take the square root of that value.
Breakdown of Standard Form Components
| Component | Description |
|---|---|
| $(x - h)^2$ | Relates to the x-coordinate of the center. |
| $(y - k)^2$ | Relates to the y-coordinate of the center. |
| $(z - l)^2$ | Relates to the z-coordinate of the center. |
| $\mathbf{r^2}$ | The square of the sphere's radius. |
Example: Standard Form
Let's find the center and radius of a sphere given the equation:
$$(x - 2)^2 + (y + 4)^2 + (z - 1)^2 = 25$$
-
Identify the Center (h, k, l):
- From $(x - 2)^2$, we have $h = 2$.
- From $(y + 4)^2$, which can be written as $(y - (-4))^2$, we have $k = -4$.
- From $(z - 1)^2$, we have $l = 1$.
- Therefore, the center of the sphere is $\mathbf{(2, -4, 1)}$.
-
Identify the Radius (r):
- The right side of the equation is $r^2 = 25$.
- To find $r$, take the square root: $r = \sqrt{25} = \mathbf{5}$.
- Thus, the radius of the sphere is $\mathbf{5}$.
Finding the Center and Radius from General Form
If the sphere's equation is not in standard form, it is typically in its general (expanded) form. To find the center and radius from this form, you must convert it to the standard form by using a technique called completing the square.
A general form equation often looks like:
$$Ax^2 + Ay^2 + Az^2 + Dx + Ey + Fz + G = 0$$
(Note: For a sphere, the coefficients of $x^2$, $y^2$, and $z^2$ must be equal.)
Steps to Complete the Square
Let's use an example to illustrate the process: Find the center and radius of the sphere given the equation:
$$x^2 + y^2 + z^2 - 6x + 2y - 4z - 11 = 0$$
-
Group Terms and Isolate the Constant:
Rearrange the equation by grouping the $x$ terms, $y$ terms, and $z$ terms together. Move the constant term to the right side of the equation.
$$(x^2 - 6x) + (y^2 + 2y) + (z^2 - 4z) = 11$$ -
Complete the Square for Each Variable:
For each grouped quadratic expression (e.g., $x^2 + Bx$), add $(\frac{\text{coefficient of linear term}}{2})^2$ to both sides of the equation.- For x-terms: The coefficient of $x$ is $-6$. Add $(\frac{-6}{2})^2 = (-3)^2 = 9$ to both sides.
- For y-terms: The coefficient of $y$ is $2$. Add $(\frac{2}{2})^2 = (1)^2 = 1$ to both sides.
- For z-terms: The coefficient of $z$ is $-4$. Add $(\frac{-4}{2})^2 = (-2)^2 = 4$ to both sides.
The equation now becomes:
$$(x^2 - 6x + 9) + (y^2 + 2y + 1) + (z^2 - 4z + 4) = 11 + 9 + 1 + 4$$ -
Factor and Simplify:
Factor each perfect square trinomial into the form $(x - h)^2$, $(y - k)^2$, $(z - l)^2$. Then, sum the constants on the right side of the equation.$$(x - 3)^2 + (y + 1)^2 + (z - 2)^2 = 25$$
-
Identify Center and Radius:
Now that the equation is in standard form, you can directly read the center coordinates $(h, k, l)$ and calculate the radius $r = \sqrt{r^2}$.-
Center:
- From $(x - 3)^2$, $h = 3$.
- From $(y + 1)^2$, $k = -1$.
- From $(z - 2)^2$, $l = 2$.
- Thus, the center of the sphere is $\mathbf{(3, -1, 2)}$.
-
Radius:
- The right side is $r^2 = 25$.
- The radius $r = \sqrt{25} = \mathbf{5}$.
-
Key Considerations
- Coefficients of Quadratic Terms: Before completing the square, ensure that the coefficients of $x^2$, $y^2$, and $z^2$ are all equal to 1. If they are not (e.g., $2x^2 + 2y^2 + 2z^2 + ...$), divide the entire equation by that common coefficient first.
- Square Root for Radius: Always remember to take the square root of the constant on the right side of the standard form equation to find the actual radius. A common mistake is to state $r^2$ as the radius.
Further Resources
For a more in-depth understanding of the completing the square technique, which is fundamental to converting the general form equation, you can explore educational resources like Khan Academy's lessons on quadratic equations.
