4. Show that x2 + y2 + z2 + 4x – 6y + 2z + 6 = 0 is the equation of a sphere. Also, find its center and radius.
x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0
x^2+4x + y^2-6y + z^2+2z = -6
x^2+4x+4 + y^2-6y+9 + z^2+2x+1 = -6+4+9+1
(x+2)^2 + (y-3)^2 + (z+1)^2 = 8
center at (-2,3,-1) and radius = √8
(-6,8/4,3) radius =8
Well, solving for humor on this one can be a bit tricky, but here's my attempt.
To show that the equation x^2 + y^2 + z^2 + 4x – 6y + 2z + 6 = 0 represents a sphere, we need to complete the square for x, y, and z.
So, let's group the terms:
(x^2 + 4x) + (y^2 - 6y) + (z^2 + 2z) = -6
To complete the square for x, we need to add (4/2)^2 = 4 to both sides:
(x^2 + 4x + 4) + (y^2 - 6y) + (z^2 + 2z) = -6 + 4
(x + 2)^2 + (y^2 - 6y) + (z^2 + 2z) = -2
Now, let's complete the square for y:
(x + 2)^2 + (y^2 - 6y + 9) + (z^2 + 2z) = -2 + 9
(x + 2)^2 + (y - 3)^2 + (z^2 + 2z) = 7
Lastly, let's complete the square for z:
(x + 2)^2 + (y - 3)^2 + (z^2 + 2z + 1) = 7 + 1
(x + 2)^2 + (y - 3)^2 + (z + 1)^2 = 8
Voila! We've successfully transformed the equation into the standard form of a sphere: (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) represents the center and r represents the radius.
Therefore, the center of the sphere is (-2, 3, -1) and the radius is sqrt(8). But hey, don't blame me if you end up running circles trying to solve this one!
To show that the given equation represents a sphere, we need to rewrite it in a standard form. The general form of the equation of a sphere is:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
where (h, k, l) represents the center of the sphere and r represents the radius.
Let's begin by completing the square for the given equation.
x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0
Rearranging the terms:
(x^2 + 4x) + (y^2 - 6y) + (z^2 + 2z) = -6
To complete the square for each variable, we need to add and subtract the square of half the coefficient of that variable. Let's perform this operation for x, y, and z separately.
For x:
x^2 + 4x + (4/2)^2 = (x + 2)^2
For y:
y^2 - 6y + (-6/2)^2 = (y - 3)^2
For z:
z^2 + 2z + (2/2)^2 = (z + 1)^2
Substituting these results back into the original equation:
(x + 2)^2 + (y - 3)^2 + (z + 1)^2 = -6 + 4 + 9 + 1 = 8
As you can see, the equation is now in the standard form:
(x - (-2))^2 + (y - 3)^2 + (z - (-1))^2 = 8
Comparing it with the general equation of a sphere, we can determine that the center is (-2, 3, -1) and the radius is √8 or 2√2.
Hence, the given equation represents a sphere with center (-2, 3, -1) and radius 2√2.