Question

Find the equation of the sphere centered at (2,−9,1) with radius 2. 4 = (x-2)+(y+9)+(z-1). Give an equation which describes the intersection of this sphere with the plane z=2.

Answer 1

The equation of the sphere centered at (2, -9, 1) with radius 2 is (x - 2)² + (y + 9)² + (z - 1)² = 4. The intersection of this sphere with the plane z=2 is a circle with the equation (x - 2)² + (y + 9)² = 3.

Step-by-step explanation:

The equation of a sphere with center (h, k, l) and radius r is given by (x - h)² + (y - k)² + (z - l)² = r². For a sphere centered at (2, -9, 1) with radius 2, the equation is (x - 2)² + (y + 9)² + (z - 1)² = 2² or (x - 2)² + (y + 9)² + (z - 1)² = 4.

To find the equation of the intersection of this sphere with the plane z = 2, substitute z = 2 into the sphere's equation to get (x - 2)² + (y + 9)² + (2 - 1)² = 4. Simplifying this gives us (x - 2)² + (y + 9)² = 3, which is the equation of a circle in the xy-plane with center (2, -9) and radius √3.

Answer 2

See explanation

Step-by-step explanation:

The equation of the shpere canterd at point
`(a,b,c)` with radius
`R` is

`(x-a)^2+(y-b)^2+(z-c)^2=R^2`

The center of the sphere is (2,-9,1), the radius is 2, then the equation is

`(x-2)^2+(y+9)^2+(z-1)^2=2^2\\ \\(x-2)^2+(y+9)^2+(z-1)^2=4`

Now, find the intersection of this sphere with the plane z=2. Substitute 2 instead of z into the sphere equation:

`(x-2)^2 +(y+9)^2+(2-1)^2=4\\ \\(x-2)^2+(y+9)^2+1=4\\ \\(x-2)^2+(y+9)^2=3`

This is the equation of the circle lying in the plane z=2 with center at (2,-9,2) and radius
`√(3).`