Find an equation for the surface consisting of all points P(x, y, z) equidistant from the point P(0, 1, 0) and the plane x = 3.

Answer Choices:
1. y2 + z2 − 6x + 2y − 8 = 0
2. x2 + z2 + 6y − 2z − 8 = 0
3. y2 + z2 + 6x − 2y − 8 = 0
4. x2 + z2 − 6y + 2z − 8 = 0
5. x2 + y2 − 2x + 6z − 8 = 0
6. x2 + y2 + 2x − 6z − 8 = 0

To find the equation for the surface consisting of all points equidistant from a given point and a given plane, we can follow these steps:

Step 1: Find the distance between the given point P(0, 1, 0) and the given plane x = 3.

The distance formula between a point (x1, y1, z1) and a plane Ax + By + Cz + D = 0 is given by:
distance = |Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)

In this case, the equation of the plane is x = 3, so A = 1, B = 0, C = 0, and D = -3. Substituting these values and the coordinates of the point P into the distance formula, we get:
distance = |1(0) + 0(1) + 0(0) + (-3)| / sqrt(1^2 + 0^2 + 0^2)
distance = 3 / 1
distance = 3

Step 2: Write the equation for the surface equidistant from P(0, 1, 0) and the plane x = 3.

Since the required surface consists of all points equidistant (distance = 3) from the given point and the given plane, the equation of the surface will be of the form:
(x - x1)^2 + (y - y1)^2 + (z - z1)^2 = distance^2

Substituting the values x1 = 0, y1 = 1, z1 = 0, and distance = 3, we get:
(x - 0)^2 + (y - 1)^2 + (z - 0)^2 = 3^2
x^2 + (y - 1)^2 + z^2 = 9

Matching this equation with the answer choices, we can see that option 3 is the correct equation:
y^2 + z^2 + 6x - 2y - 8 = 0

So, the correct equation for the surface is y^2 + z^2 + 6x - 2y - 8 = 0 (Option 3).

To find an equation for the surface consisting of all points equidistant from the point P(0, 1, 0) and the plane x = 3, we need to consider the distance formula.

The distance between a point P(x, y, z) and another point Q(a, b, c) is given by the equation:
distance(PQ) = √[(x - a)² + (y - b)² + (z - c)²]

Using this formula, let's find the distance between P(x, y, z) and P(0, 1, 0):
distance(P(0, 1, 0)P(x, y, z)) = √[(x - 0)² + (y - 1)² + (z - 0)²]
distance(P(0, 1, 0)P(x, y, z)) = √[x² + (y - 1)² + z²]

Now let's find the distance between P(x, y, z) and the plane x = 3:
distance(P(x, y, z)(3, y, z)) = √[(x - 3)² + (y - y)² + (z - z)²]
distance(P(x, y, z)(3, y, z)) = √[(x - 3)²]

Since we want all points to be equidistant from P(0, 1, 0) and the plane x = 3, the distances we calculated should be the same. Therefore, we have:

√[x² + (y - 1)² + z²] = √[(x - 3)²]

Square both sides of the equation:
x² + (y - 1)² + z² = (x - 3)²

Expand the squared terms:
x² + y² - 2y + 1 + z² = x² - 6x + 9

Simplify and rearrange the equation:
y² + z² - 2y + 1 - 6x + 9 = 0
y² + z² - 2y - 6x + 10 = 0

Comparing this equation to the answer choices, we see that:
Answer Choice 1. y² + z² - 6x + 2y - 8 = 0

Therefore, the correct equation for the surface consisting of all points P(x, y, z) equidistant from the point P(0, 1, 0) and the plane x = 3 is option 1.