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 , , , and , we start by writing the general equation of a sphere:
Here, , , , and 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
Substitute into the general equation:
This simplifies to:
Thus, the equation becomes:
Step 2: Use the point
Substitute into the equation:
This simplifies to:
Thus, .
Step 3: Use the point
Substitute into the equation:
This simplifies to:
Thus, .
Step 4: Use the point
Substitute into the equation:
This simplifies to:
Thus, .
Step 5: Write the final equation
Now, substitute , , and into the equation:
Thus, the equation of the sphere is:
Final Answer:
The equation of the sphere passing through the points , , , and is:
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:
- How can we find the center and radius of a sphere given its equation?
- What is the general form of the equation of a sphere in 3D space?
- How do we determine the equation of a sphere that passes through a set of points in space?
- Can the equation of a sphere be written in terms of its center and radius?
- 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
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