Answer: the equation for the surface consisting of all points equidistant from the point (-3, 0, 0) and the plane x = 3 is simply x = 0.
Explanation:
To find an equation for the surface consisting of all points equidistant from the point (-3, 0, 0) and the plane x = 3, we can start by considering the distance between a generic point (x, y, z) on the surface and the given point.
The distance between two points in 3D space can be calculated using the distance formula:
d = sqrt((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Here, (x₁, y₁, z₁) represents the coordinates of the given point (-3, 0, 0), and (x₂, y₂, z₂) represents the coordinates of a generic point on the surface.
Since we want all points on the surface to be equidistant from (-3, 0, 0) and the plane x = 3, the distances should be equal. Therefore, we can set up the following equation:
sqrt((x - (-3))² + (y - 0)² + (z - 0)²) = sqrt((x - 3)² + (y - 0)² + (z - 0)²)
Simplifying this equation, we get:
sqrt((x + 3)² + y² + z²) = sqrt((x - 3)² + y² + z²)
Squaring both sides of the equation to eliminate the square root, we have:
(x + 3)² + y² + z² = (x - 3)² + y² + z²
Expanding and simplifying further, we get:
x² + 6x + 9 + y² + z² = x² - 6x + 9 + y² + z²
The terms y² and z² cancel out, and we are left with:
6x + 9 = -6x + 9
Simplifying, we have:
12x = 0
Dividing both sides by 12, we find:
x = 0