Find the length of perpendicular from the point (1,1,2) to the plane `2x – 2y + 4z + 5 = 0`.

Options

A. `(13sqrt6)/(5)` units
B. `(13sqrt6)/(12)` units
C. `(5sqrt3)/(6)` units
D. `(13sqrt5)/(12)` units

Answer

To find the length of the perpendicular from the point \( P(1, 1, 2) \) to the plane given by the equation \( 2x - 2y + 4z + 5 = 0 \), we can use the formula for the distance \( d \) from a point \( (x_1, y_1, z_1) \) to the plane \( ax + by + cz + d = 0 \):

\[ d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

### Step-by-step Solution:

1. **Identify the coefficients from the plane equation**:
The plane equation is \( 2x - 2y + 4z + 5 = 0 \). Here, we can identify:
- \( a = 2 \)
- \( b = -2 \)
- \( c = 4 \)
- \( d = 5 \)

2. **Substitute the coordinates of the point into the formula**:
The coordinates of the point are \( (x_1, y_1, z_1) = (1, 1, 2) \). We substitute these values into the distance formula:
\[ d = \frac{|2(1) + (-2)(1) + 4(2) + 5|}{\sqrt{2^2 + (-2)^2 + 4^2}} \]

3. **Calculate the numerator**:
Now, calculate the expression in the numerator:
\[ 2(1) + (-2)(1) + 4(2) + 5 = 2 - 2 + 8 + 5 = 13 \]
So, the numerator becomes \( |13| = 13 \).

4. **Calculate the denominator**:
Next, calculate the expression in the denominator:
\[ \sqrt{2^2 + (-2)^2 + 4^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6} \]

5. **Combine the results**:
Now, substitute the values back into the distance formula:
\[ d = \frac{13}{2\sqrt{6}} \]

6. **Rationalize the denominator**:
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{6} \):
\[ d = \frac{13\sqrt{6}}{2 \cdot 6} = \frac{13\sqrt{6}}{12} \]

### Final Answer:
The length of the perpendicular from the point \( (1, 1, 2) \) to the plane \( 2x - 2y + 4z + 5 = 0 \) is:
\[ \frac{13\sqrt{6}}{12} \text{ units} \]