Understanding Spheres: Equations and Properties

This video tutorial covers solving problems related to spheres. It begins with finding the equation of a sphere given its center and radius, using the standard formula. The tutorial then explains how to derive the equation of a sphere when given the endpoints of its diameter, by calculating the midpoint and using the distance formula to find the radius. Finally, it demonstrates how to complete the square to rewrite a given sphere equation in standard form, allowing for easy identification of the sphere's center and radius.

What is the standard form of the equation of a sphere with center (h, k, l) and radius r?

• (x + h)^2 + (y + k)^2 + (z + l)^2 = r^2

• x^2 + y^2 + z^2 = r^2

• (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

• x^2 + y^2 + z^2 + 2hx + 2ky + 2lz = r^2

If a sphere has a center at (2, -4, 3) and a radius of 5, what is the equation of the sphere?

• (x - 2)^2 + (y + 4)^2 + (z - 3)^2 = 25

• (x + 2)^2 + (y + 4)^2 + (z + 3)^2 = 5

• (x + 2)^2 + (y - 4)^2 + (z + 3)^2 = 25

• (x - 2)^2 + (y - 4)^2 + (z - 3)^2 = 5

How do you find the center of a sphere given the endpoints of its diameter?

• By multiplying the coordinates of the endpoints

• By adding the coordinates of the endpoints

• By averaging the x, y, and z coordinates of the endpoints

• By subtracting the coordinates of one endpoint from the other

Given endpoints (1, -6, -5) and (5, 2, 7), what is the center of the sphere?

• (4, 1, 3)

• (2, -4, 6)

• (3, -2, 1)

• (0, -3, 2)

What is the formula to calculate the radius of a sphere from its center and a point on the sphere?

• The sum of the squares of the differences in coordinates

• The average of the differences in coordinates

• The square root of the sum of the squares of the differences in coordinates

• The product of the differences in coordinates

What is the radius of a sphere with center (3, -2, 1) and a point (5, 2, 7) on its surface?

• √50

• √56

• √60

• √64

How do you find the center of a sphere from its equation by completing the square?

• By isolating the x, y, and z terms

• By factoring the equation

• By grouping and completing the square for each variable

• By differentiating the equation

What is the center of the sphere given by the equation x^2 + y^2 + z^2 - 4x + 6y + 10z = -29?

• (2, -3, -5)

• (4, -6, -10)

• (1, 3, 5)

• (0, 0, 0)

What is the radius of the sphere with the equation x^2 + y^2 + z^2 - 4x + 6y + 10z = -29 after completing the square?

• 3

• 4

• 2

• 5

What is the purpose of completing the square in the context of finding a sphere's equation?

• To simplify the equation

• To find the center and radius of the sphere

• To convert the equation to a linear form

• To eliminate the z variable