The shortest distance from the plane `12 x+4y+3z=327` to the sphere `x^2+y^2+z^2+4x-2y-6z=155` is a. `39` b. `26` c. `41-4/(13)` d. `13`

Options

A. 39
B. 26
C. `41(4)/(13)`
D. 13

Explanation

To find the shortest distance from the plane \(12x + 4y + 3z = 327\) to the sphere \(x^2 + y^2 + z^2 + 4x - 2y - 6z = 155\), we can follow these steps:

### Step 1: Find the center and radius of the sphere
The equation of the sphere can be rewritten in standard form. We start with:
\[ x^2 + y^2 + z^2 + 4x - 2y - 6z = 155 \]
We complete the square for each variable:
...