Write the equation of the sphere in standard form.
x2 + y2 + z2 + 4x − 2y − 2z = 19
Just needed a refresher. Thanks.
To write the equation of the sphere in standard form, the equation should be in the form:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
where (h, k, l) are the coordinates of the center of the sphere and r is the radius.
To rearrange the given equation into this form, let's complete the square for each variable.
x^2 + y^2 + z^2 + 4x - 2y - 2z = 19
Rearrange the equation by grouping the x, y, and z terms separately:
(x^2 + 4x) + (y^2 - 2y) + (z^2 - 2z) = 19
Now, complete the square for each variable:
(x^2 + 4x + 4) + (y^2 - 2y + 1) + (z^2 - 2z + 1) = 19 + 4 + 1 + 1
(x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 25
Therefore, the equation of the sphere in standard form is:
(x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 25
To write the equation of the sphere in standard form, we need to complete the square for the x, y, and z variables.
First, let's group the like terms:
(x^2 + 4x) + (y^2 - 2y) + (z^2 - 2z) = 19
Now we need to complete the square for each variable separately.
For the x-term, we need to add the square of half the coefficient of x, which is (4/2)^2 = 4:
(x^2 + 4x + 4) + (y^2 - 2y) + (z^2 - 2z) = 19 + 4
Simplifying the x-term:
(x + 2)^2 + (y^2 - 2y) + (z^2 - 2z) = 23
Similarly, for the y-term, we add the square of half the coefficient of y, which is (-2/2)^2 = 1:
(x + 2)^2 + (y^2 - 2y + 1) + (z^2 - 2z) = 23 + 1
Simplifying the y-term:
(x + 2)^2 + (y - 1)^2 + (z^2 - 2z) = 24
Lastly, for the z-term, we add the square of half the coefficient of z, which is (-2/2)^2 = 1:
(x + 2)^2 + (y - 1)^2 + (z^2 - 2z + 1) = 24 + 1
Simplifying the z-term:
(x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 25
Therefore, the equation of the sphere in standard form is:
(x + 2)^2 + (y - 1)^2 + (z - 1)^2 = 25
come on - you did this for 2D figures back in Algebra II:
x^2+4x + y^2-2y + z^2-2z = 19
x^2+4x+4 + y^2-2y+1 = z^2-2z+1 = 19+4+1+1
(x+2)^2 + (y-1)^2 + (z-1)^2 = 25