Use the midpoint formula to find the equations of the spheres in Exercises 36 and 37 . the spher

Use the midpoint formula to find the equations of the...

Key Concepts

Diameter-Radius Relationship

A key geometric concept for spheres is that the midpoint of a diameter is the center of the sphere, and the radius is exactly half the length of the diameter. This relationship is used to transition from knowing the endpoints of a diameter to obtaining the center and radius necessary for writing the sphere's equation.

Sphere Equation

The equation of a sphere in three-dimensional space is typically expressed as (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, where (a, b, c) represents the center and r the radius of the sphere. This formula encapsulates the set of points that are at a constant distance from the center, which is foundational in solving problems involving spheres.

Midpoint Formula

The midpoint formula is used to determine the exact middle point between two points in space by averaging their corresponding coordinates. In coordinate geometry, this formula is essential to locating the center of a line segment, which in the context of spheres, gives the center when the segment is a diameter.

Distance Formula

The distance formula calculates the straight-line distance between two points in a coordinate system. It is crucial for determining the length of the diameter of a sphere. Once the diameter is known via the distance between the endpoints, the radius of the sphere can be easily computed.

Transcript

00:01 For this problem, we are asked to use the midpoint formula to find the equations of the sphere in which the segment with endpoints 3 -2 -6 and 5 -6 -4 is a diameter.

00:09 So, in that case, that tells us that 3 minus x -not squared plus negative 2 minus y -not squared plus 6 minus z -not squared must be equal to 5 minus x -0.

00:32 Squared plus 6 minus y not squared plus 4 minus z not squared they both must be satisfying the equation of the sphere so in that case we have 3 minus x not squared must be equal to 5 minus x not squared in which case the absolute value of 3 minus x not must be equal to the absolute value of 5 minus x not from taking the square root of both sides which would then mean that essentially we need 3 minus x not to be equal to 5 minus x not.

01:12 Therefore, if we, or excuse me, let me correct myself here, 3 minus x not must be equal to plus or minus 5 minus x not, which means that one possibility, which doesn't lead to something which can't be true, would be that 3 minus x not must be equal to x not minus 5.

01:31 Adding x not to both sides gives us that 2x0, or excuse me, adding x not to both sides and adding 5 to both sides gives us that 2x0 must be equal to 3 plus 5, so 2x0 must be equal to 4, and therefore x not must be equal to 4.

01:49 Sorry, i'm getting ahead of myself here.

01:51 3 plus 5 is 8, which then means that x not must be equal to 4.

01:56 And we can see that both x equals 3 and x equals 5 are the same distance from x not.

02:03 Applying a similar procedure for everything else we would have that negative 2 minus y not must be equal to y not minus 6 again if we used the if we used the positive absolute values were equal to each other then we would run into the problem of a trivial statement which can't be true so in this case we add why not to both sides we get two y not would be equal to add six to both sides to why not equals 4.

02:34 And so y not equals 2.

02:36 That doesn't make sense...

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