Equations of Spheres

​Multidimensional Coordinate Systems

Replace this with a sub-header that can be in multiple lines. ​

Review from Yesterday

• When plotting a point in space, we need a 3rd Axis, the z-axis

• Because of this, any point in space would have the coordinate of (x,y,z) ALWAYS IN THIS ORDER!!

• We can use the same midpoint and distance formulas from Algebra for these 3D points, just with the z-portion added on

Equation of a Sphere

(x-h)² + (y-k)² + (z-l)² = r²

​

Center located at ( h, k, l ) with Radius r.

Example 1

• What would be the equation of the Sphere with a Center at (1, -2, 4) and a Radius of 2?

• Plug the h, k, l, and r into the Sphere Equation: (x - h)² + (y - k)² + (z - l)² = r²

• (x - 1)² + (y - -2)² + (z - 4)² = 2²

• Simplify (if needed)

• (x - 1)² + (y + 2)² + (z - 4)² = 4

Multiple Choice

Example 2: Which of the following would be the correct Equation for a Sphere with a Center at (-1, 0, 3) and a Radius of 10?

• (x + 1)² + y² + (z - 3)² = 100

• (x - 1)² + y² + (z + 3)² = 100

• (z + 1)² - (x + 3)² - y² = 100

Multiple Choice

Example 3: What would be the equation for a Circle with a center of (8,3,7) and a Radius of 14?

• (x - 3)² + (y - 8)² + (z - 7)² = 196

• (x - 8)² + (y - 3)² + (z - 7)² = 196

• (x - 4)² + (y - 1.5)² + (z - 3.5)² = 196

​What if it isn't in Standard Form?

​If you are given a Sphere equation that ISN'T in Standard Form, to convert it to that, you'll need to go through the steps of Completing the Square, for ALL 3 VARIABLES!!!

​

The next slide has a video example going through this to help remind you of the steps for Completing the Square...​

​Converting an Equation to Standard Form

​

​Converting an Equation to Standard Form (2nd Ex)

​

Multiple Choice

Now you give it a Try:

Convert to the Standard Form of a Sphere:

$$x^2+y^2+z^2-8x+10y+20z-8=0$$  

• $$\left(x-4\right)^2+\left(y+5\right)^2+\left(z+10\right)^2=149$$  

• $$\left(x-8\right)^2+\left(y+10\right)^2+\left(z+20\right)^2=149$$  

• $$\left(x-8\right)^2+\left(y+10\right)^2+\left(z+20\right)^2=8$$  

• $$\left(x-4\right)^2+\left(y+5\right)^2+\left(z+10\right)^2=8$$  

Multiple Choice

Example 2: Convert to Standard Form

$$y^2+x^2+z^2+5y-7x+8z=4$$  

• $$\left(x-3.5\right)^2+\left(y+2.5\right)^2+\left(z+4\right)^2=14$$  

• $$\left(x-3.5\right)^2+\left(y+2.5\right)^2+\left(z+4\right)^2=38.5$$  

• $$\left(x-3.5\right)^2+\left(y+2.5\right)^2+\left(z+4\right)^2=4$$  

• $$\left(x-7\right)^2+\left(y+5\right)^2+\left(z+8\right)^2=38.5$$  

Draw

Summary: Draw a Cat