Math Problem Statement

find the equations of the sphere which passes through the points (0,0,0) , (1,0,0),(0,1,0),and (0,0,1)

Solution

To find the equation of the sphere that passes through the points (0,0,0)(0,0,0), (1,0,0)(1,0,0), (0,1,0)(0,1,0), and (0,0,1)(0,0,1), we start by writing the general equation of a sphere:

x2+y2+z2+Dx+Ey+Fz+G=0x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0

Here, DD, EE, FF, and GG are constants to be determined. We use the fact that the sphere passes through the given points, which will give us a system of equations.

Step 1: Use the point (0,0,0)(0, 0, 0)

Substitute (x,y,z)=(0,0,0)(x, y, z) = (0, 0, 0) into the general equation:

02+02+02+D(0)+E(0)+F(0)+G=00^2 + 0^2 + 0^2 + D(0) + E(0) + F(0) + G = 0

This simplifies to:

G=0G = 0

Thus, the equation becomes:

x2+y2+z2+Dx+Ey+Fz=0x^2 + y^2 + z^2 + Dx + Ey + Fz = 0

Step 2: Use the point (1,0,0)(1, 0, 0)

Substitute (x,y,z)=(1,0,0)(x, y, z) = (1, 0, 0) into the equation:

12+02+02+D(1)+E(0)+F(0)=01^2 + 0^2 + 0^2 + D(1) + E(0) + F(0) = 0

This simplifies to:

1+D=01 + D = 0

Thus, D=1D = -1.

Step 3: Use the point (0,1,0)(0, 1, 0)

Substitute (x,y,z)=(0,1,0)(x, y, z) = (0, 1, 0) into the equation:

02+12+02+D(0)+E(1)+F(0)=00^2 + 1^2 + 0^2 + D(0) + E(1) + F(0) = 0

This simplifies to:

1+E=01 + E = 0

Thus, E=1E = -1.

Step 4: Use the point (0,0,1)(0, 0, 1)

Substitute (x,y,z)=(0,0,1)(x, y, z) = (0, 0, 1) into the equation:

02+02+12+D(0)+E(0)+F(1)=00^2 + 0^2 + 1^2 + D(0) + E(0) + F(1) = 0

This simplifies to:

1+F=01 + F = 0

Thus, F=1F = -1.

Step 5: Write the final equation

Now, substitute D=1D = -1, E=1E = -1, and F=1F = -1 into the equation:

x2+y2+z2xyz=0x^2 + y^2 + z^2 - x - y - z = 0

Thus, the equation of the sphere is:

x2+y2+z2xyz=0x^2 + y^2 + z^2 - x - y - z = 0

Final Answer:

The equation of the sphere passing through the points (0,0,0)(0, 0, 0), (1,0,0)(1, 0, 0), (0,1,0)(0, 1, 0), and (0,0,1)(0, 0, 1) is:

x2+y2+z2xyz=0x^2 + y^2 + z^2 - x - y - z = 0


Would you like further details or explanations on any step? Let me know if you have any questions.

Here are some related questions you may find interesting:

  1. How can we find the center and radius of a sphere given its equation?
  2. What is the general form of the equation of a sphere in 3D space?
  3. How do we determine the equation of a sphere that passes through a set of points in space?
  4. Can the equation of a sphere be written in terms of its center and radius?
  5. How do we derive the equation of a sphere from its general form to standard form?

Tip: When solving for the constants in the equation of a sphere, substituting the coordinates of given points helps eliminate the unknowns systematically!

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Equation of a Sphere
Systems of Equations

Formulas

General equation of a sphere: x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0

Theorems

Equations of a sphere
Substitution method

Suitable Grade Level

Grades 10-12