Answer:
the correct answer is:
c) Radius = 2, Center = (4, 2, 2)
Step-by-step explanation:
To find the radius and center of the sphere given by the equation [tex]x^{2} +y^{2} +z^{2} - 8x - 4y- 4z = -20,[/tex]we need to complete the square for each variable. The standard form for the equation of a sphere is [tex](x-h)^{2} +(y-k)^{2} (z-l)^{2} =r^{2}[/tex], where (h,k,l) is the center of the sphere, and r is the radius.
Starting with the given equation:
[tex]x^{2} +y^{2} +z^{2} - 8x - 4y- 4z = -20,[/tex]
Now, complete the square for
x2−8x+?+y2−4y+?+z2−4z+?=−20+?+?+?
To complete the square for x, we need to add (8/2)=16 inside the parentheses:
x2−8x+16+y2−4y+?+z2−4z+?=−20+16+?+?
Similarly, complete the square for y and z:
x2−8x+16+y2−4y+4+z2−4z+4=−20+16+4+4
Now, factor the perfect squares:
[tex](x-4)^{2} +(y-2)^{2} +(z-2)^{2} =4[/tex]
Now, the equation is in the standard form. The center of the sphere is
(h,k,l)=(4,2,2), and the radius is r = [tex]\sqrt{4}[/tex] = 2
So, the correct answer is:
c) Radius = 2, Center = (4, 2, 2)
The center of the sphere is (4, 2, 2) and the radius is 5. The Correct Answer is Option.A.
To find the radius and center of the sphere given by the equation (x² + y² + z² - 8x - 4y - 4z = -20), we can rewrite the equation in the form (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center of the sphere and r is the radius.
Comparing the given equation to this form, we have (x - 4)² + (y - 2)² + (z - 2)² = 5²,
so the center of the sphere is (4, 2, 2) and the radius is 5.
