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

Answer:

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 $\sqrt{3}.$