The plane $2x-2y+z+12=0$ touches the sphere $x^{2}+y^{2}+z^{2}-2x-4y+2z-3=0$ at the point:

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to verify that the plane $2x - 2y + z + 12 = 0$ is tangent to the sphere $x^2 + y^2 + z^2 - 2x - 4y + 2z - 3 = 0$ at the point $(-1, 4, -2)$.

### Explanation:

1. **Equation of the Sphere:**
   The given sphere equation is:
   $$
   x^2 + y^2 + z^2 - 2x - 4y + 2z - 3 = 0
   $$
   This can be rewritten in the standard form by completing the square:
   $$
   (x^2 - 2x) + (y^2 - 4y) + (z^2 + 2z) = 3
   $$
   Completing the square for each variable:
   $$
   (x-1)^2 - 1 + (y-2)^2 - 4 + (z+1)^2 - 1 = 3
   $$
   $$
   (x-1)^2 + (y-2)^2 + (z+1)^2 = 9
   $$
   This represents a sphere with center $(1, 2, -1)$ and radius 3.

2. **Equation of the Plane:**
   The plane is given by:
   $$
   2x - 2y + z + 12 = 0
   $$

3. **Point of Tangency:**
   For the plane to be tangent to the sphere at a point, the point must satisfy both the sphere's equation and the plane's equation.

- **Checking Option 4 $(-1, 4, -2)$:**
     - Substitute into the sphere's equation:
       $$
       (-1)^2 + 4^2 + (-2)^2 - 2(-1) - 4(4) + 2(-2) - 3 = 1 + 16 + 4 + 2 - 16 - 4 - 3 = 0
       $$
       The point satisfies the sphere's equation.
     - Substitute into the plane's equation:
       $$
       2(-1) - 2(4) + (-2) + 12 = -2 - 8 - 2 + 12 = 0
       $$
       The point satisfies the plane's equation.

Since $(-1, 4, -2)$ satisfies both the sphere and the plane equations, it is the point of tangency.

4. **Why Other Options are Incorrect:**
   - **Option 1 $(1, -2, 3)$:**
     - Substituting into the sphere's equation does not yield zero.
   - **Option 2 $(1, 2, -1)$:**
     - Although this point is the center of the sphere, it does not satisfy the plane's equation.
   - **Option 3 $(-1, -4, 3)$:**
     - Substituting into the sphere's equation does not yield zero.

Thus, Option 4 $(-1, 4, -2)$ is the only point that satisfies both the sphere and the plane equations, confirming it as the correct point of tangency.