Question

Write an equation of a sphere with center ( 0 , 8 , 0 ) (0,8,0) , and with the sphere tangent to the plane with equation x

Answer 1

The equation of the sphere with center (0, 8, 0) that is tangent to the plane x=0 is simply x² + (y - 8)² + z² = 0.

Step-by-step explanation:

To write the equation of a sphere with center at (0, 8, 0), we need to determine its radius. The sphere is tangent to the plane with equation x, which means the distance from the center of the sphere to the plane x=0 (the yz-plane) is equal to the radius of the sphere. Since the center of the sphere is at x=0 already, the sphere is tangent to the yz-plane at the center. Thus, the radius of the sphere is 0, and the equation of a sphere is given by (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. The equation of the sphere with center (0, 8, 0) and radius 0 is:

x² + (y - 8)² + z² = 0.

Answer 2

`\mathbf{ x^2 + (y-8)^2 +(z)^2 = 32}`

Step-by-step explanation:

Given that:

The center of the sphere = (0,8,0)

`The \ equation \ of \ the \ tangent \ plane \ is:`

= x - y = 0

x = y

However, the radius of the sphere is equal to the perpendicular distance of the plane from the center.

As such, the radius
`r = \Big | (ax_1 +by_1+cz_1)/(√(a^2 + b^2 +c^2)) \Big |`

`r = \Big | (1*0 +(-1)*8+0*0)/(√(1^2 + (-1)^2 +0^2)) \Big |`

`r = \Big | (-8)/(√(2)) \Big |`

`r = (8)/(√(2))`

`r = (4 * 2)/(√(2))`

`r = 4√(2)` units

Thus, the equation of the sphere
`r^2 = (x -x_1)^2 +(y-y_1)^2 +(z -z_1)^2`

`\mathbf{\implies x^2 + (y-8)^2 +(z)^2 = 32}`