The point in which the line `(x-2)/(3)=(y+1)/(4)=(z-2)/(12)` meets the plane `x-2y+z=20` is

Options

A. `(7, -8, 26)`
B. `(8, 7, 26)`
C. `(7, 8, 26)`
D. none of these

Answer

To find the point where the line \(\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}\) meets the plane \(x - 2y + z = 20\), we can follow these steps:

### Step 1: Parametrize the Line
The given line can be expressed in parametric form. Let \(t\) be the parameter:
\[ x = 3t + 2 \]
\[ y = 4t - 1 \]
\[ z = 12t + 2 \]

### Step 2: Substitute into the Plane Equation
Now, substitute \(x\), \(y\), and \(z\) into the plane equation \(x - 2y + z = 20\):
\[ (3t + 2) - 2(4t - 1) + (12t + 2) = 20 \]

### Step 3: Simplify the Equation
Expanding the equation:
\[ 3t + 2 - 8t + 2 + 12t + 2 = 20 \]
Combine like terms:
\[ (3t - 8t + 12t) + (2 + 2 + 2) = 20 \]
\[ 7t + 6 = 20 \]

### Step 4: Solve for \(t\)
Now, isolate \(t\):
\[ 7t = 20 - 6 \]
\[ 7t = 14 \]
\[ t = 2 \]

### Step 5: Find the Coordinates
Now substitute \(t = 2\) back into the parametric equations to find the coordinates:
\[ x = 3(2) + 2 = 6 + 2 = 8 \]
\[ y = 4(2) - 1 = 8 - 1 = 7 \]
\[ z = 12(2) + 2 = 24 + 2 = 26 \]

### Conclusion
The point where the line meets the plane is \((8, 7, 26)\).

---