The perpendicular distance from the point (1,-1) to the line `x+5y-9=0` is equal to

Options

A. `sqrt(2/13)`
B. `sqrt(13)/2`
C. `2/13`
D. `13/2`

Answer

To find the perpendicular distance from the point (1, -1) to the line given by the equation \(x + 5y - 9 = 0\), we can use the formula for the distance \(D\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\):

\[ D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

### Step-by-Step Solution:

1. **Identify the coefficients from the line equation**:
The line equation is \(x + 5y - 9 = 0\). Here, we can identify:
- \(A = 1\)
- \(B = 5\)
- \(C = -9\)

2. **Identify the point coordinates**:
The point given is \((x_1, y_1) = (1, -1)\).

3. **Substitute the values into the distance formula**:
Now we substitute \(A\), \(B\), \(C\), \(x_1\), and \(y_1\) into the distance formula:
\[ D = \frac{|1 \cdot 1 + 5 \cdot (-1) - 9|}{\sqrt{1^2 + 5^2}} \]

4. **Calculate the numerator**:
Calculate \(1 \cdot 1 + 5 \cdot (-1) - 9\):
\[ = 1 - 5 - 9 = -13 \]
Taking the absolute value:
\[ | -13 | = 13 \]

5. **Calculate the denominator**:
Calculate \(\sqrt{1^2 + 5^2}\):
\[ = \sqrt{1 + 25} = \sqrt{26} \]

6. **Combine the results**:
Now, substitute back into the formula:
\[ D = \frac{13}{\sqrt{26}} \]

7. **Rationalize the denominator** (optional):
To rationalize, multiply the numerator and denominator by \(\sqrt{26}\):
\[ D = \frac{13 \sqrt{26}}{26} = \frac{13}{2\sqrt{13}} \]

8. **Final result**:
Thus, the perpendicular distance from the point (1, -1) to the line \(x + 5y - 9 = 0\) is:
\[ D = \frac{13}{2\sqrt{13}} \]