Understanding the Equation of a Sphere

This video tutorial explains how to write the equation of a sphere in standard form and identify its center and radius. It begins with a definition of a sphere in three-dimensional space and provides the general equation. The tutorial then walks through two example problems, demonstrating how to rewrite equations in standard form, complete the square, and solve for the center and radius, even when coefficients are present on squared terms.

What is the definition of a sphere in three-dimensional space?

• A set of points in R2 equidistant from a center

• A set of points in R3 equidistant from a center

• A set of points in R3 with varying distances from a center

• A set of points in R2 with varying distances from a center

In the standard form of a sphere's equation, what does the term (x-a)^2 represent?

• The radius of the sphere

• The x-coordinate of the center

• The y-coordinate of the center

• The z-coordinate of the center

What is the first step in rewriting a sphere's equation into standard form?

• Adding the constant term to both sides

• Multiplying all terms by a constant

• Subtracting the constant term from both sides

• Grouping the x, y, and z terms together

When completing the square, what must be done after adding a constant to one side of the equation?

• Divide the other side by the constant

• Subtract the same constant from the other side

• Add the same constant to the other side

• Multiply the other side by the constant

How do you determine the radius from the equation of a sphere in standard form?

• By squaring the constant on the right side

• By taking the square root of the constant on the right side

• By doubling the constant on the right side

• By halving the constant on the right side

In the example with coefficients, what is the purpose of factoring out a number from the squared terms?

• To simplify the equation

• To change the center of the sphere

• To make the leading coefficient 1

• To eliminate the constant term

What is the center of the sphere in the example with coefficients after simplification?

• (-1, -2, 0)

• (1, 2, 0)

• (-1, 2, 0)

• (1, -2, 0)

How is the radius of the sphere expressed in the final example?

• As the square root of 21 divided by 2

• As the square root of 21 minus 2

• As the square root of 21 multiplied by 2

• As the square root of 21 plus 2

Why is only the positive value of the radius used in the final example?

• Because the sphere is in the first quadrant

• Because the center is positive

• Because the equation requires it

• Because the radius cannot be negative

What is the final step in simplifying the equation of a sphere with coefficients?

• Multiplying all terms by the coefficient

• Dividing all terms by the coefficient

• Adding the coefficient to all terms

• Subtracting the coefficient from all terms