It seems there might be a misunderstanding in the interpretation of the problem statement. From the information provided, we are given a center at (1, -2, 4) and a radius of 3. These values are more consistent with the parameters for defining a sphere rather than a plane.
To find the equation of the sphere with center [tex]\((1, -2, 4)\)[/tex] and radius [tex]\(3\)[/tex], we follow these steps:
1. Identify the formula for the equation of a sphere:
The general equation for a sphere centered at [tex]\((X_c, Y_c, Z_c)\)[/tex] with radius [tex]\(R\)[/tex] is given by:
[tex]\[
(x - X_c)^2 + (y - Y_c)^2 + (z - Z_c)^2 = R^2
\][/tex]
2. Substitute the given center coordinates and radius into the formula:
- Center [tex]\((X_c, Y_c, Z_c) = (1, -2, 4)\)[/tex]
- Radius [tex]\(R = 3\)[/tex]
Substituting these values into the sphere equation we get:
[tex]\[
(x - 1)^2 + (y - (-2))^2 + (z - 4)^2 = 3^2
\][/tex]
3. Simplify the radius squared:
[tex]\[
3^2 = 9
\][/tex]
4. Write the final equation:
Therefore, the equation of the sphere is:
[tex]\[
(x - 1)^2 + (y + 2)^2 + (z - 4)^2 = 9
\][/tex]
Thus, the equation of the sphere with center at [tex]\((1, -2, 4)\)[/tex] and radius [tex]\(3\)[/tex] is:
[tex]\[
(x - 1)^2 + (y + 2)^2 + (z - 4)^2 = 9
\][/tex]
