Understanding the Equation of a Sphere

The video tutorial explains how to find the center and radius of a sphere by rewriting its equation in a specific form. The process involves completing the square for each variable term, adjusting the equation accordingly, and identifying the center and radius from the rewritten equation. The tutorial provides a step-by-step guide to completing the square and forming perfect squares, ultimately leading to the solution.

What is the general form of the equation of a sphere?

• x^2 + y^2 + z^2 = r^2

• (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

• (x - h)^2 + (y - k)^2 = r^2

• x^2 + y^2 = r^2

What is the first step in completing the square for the x term?

• Add 4 to both sides

• Subtract 4x and group terms

• Multiply by 2

• Divide by 2

How do you determine the number to add inside the parentheses when completing the square for y?

• Add the coefficient of y

• Double the coefficient of y

• Square the coefficient of y

• Take half of the coefficient of y and square it

What is the perfect square form of x^2 - 4x + 4?

• (x - 2)^2

• (x + 2)^2

• (x - 4)^2

• (x + 4)^2

After completing the square, what is the next step to find the center of the sphere?

• Multiply the constants

• Use the opposite signs of the numbers in the parentheses

• Identify the coefficients of x, y, and z

• Add all the constants

What is the center of the sphere given by the equation (x - 2)^2 + (y + 4)^2 + (z + 6)^2 = 61?

• (-2, -4, -6)

• (2, 4, 6)

• (-2, 4, 6)

• (2, -4, -6)

How do you find the radius of the sphere from the equation (x - 2)^2 + (y + 4)^2 + (z + 6)^2 = 61?

• Multiply 61 by 2

• Subtract 61 from 100

• Divide 61 by 2

• Take the square root of 61

What is the radius of the sphere in the equation (x - 2)^2 + (y + 4)^2 + (z + 6)^2 = 61?

• 7

• 10

• 8

• √61

Why is it necessary to add the same numbers to both sides of the equation when completing the square?

• To make the equation more complex

• To keep the equation balanced

• To simplify the equation

• To change the center of the sphere

What is the final form of the equation after completing the square and simplifying?

• (x - 2)^2 + (y + 4)^2 + (z + 6)^2 = 61

• (x + 2)^2 + (y - 4)^2 + (z - 6)^2 = 61

• x^2 + y^2 + z^2 = 61

• (x - 2)^2 + (y - 4)^2 + (z - 6)^2 = 61