Find the length of the perpendicular from the point (3,-2) to the straight line 12x - 5y + 6 = 0:

Options

A. 5
B. 4
C. 6
D. 8

Answer

To find the length of the perpendicular from the point (3, -2) to the straight line given by the equation \(12x - 5y + 6 = 0\), we will use the formula for the perpendicular distance from a point to a line.

### Step-by-Step Solution:

1. **Identify the coefficients from the line equation**:
The line equation is given as \(12x - 5y + 6 = 0\). Here, we can identify:
- \(a = 12\)
- \(b = -5\)
- \(c = 6\)

2. **Identify the coordinates of the point**:
The point from which we want to find the perpendicular distance is \((x_1, y_1) = (3, -2)\).

3. **Use the formula for the perpendicular distance**:
The formula for the perpendicular distance \(d\) from a point \((x_1, y_1)\) to the line \(ax + by + c = 0\) is given by:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

4. **Substitute the values into the formula**:
Substitute \(a\), \(b\), \(c\), \(x_1\), and \(y_1\) into the formula:
\[ d = \frac{|12(3) + (-5)(-2) + 6|}{\sqrt{12^2 + (-5)^2}} \]

5. **Calculate the numerator**:
Calculate \(12(3) + (-5)(-2) + 6\):
\[ = 36 + 10 + 6 = 52 \]
So, the numerator becomes \(|52| = 52\).

6. **Calculate the denominator**:
Calculate \(\sqrt{12^2 + (-5)^2}\):
\[ = \sqrt{144 + 25} = \sqrt{169} = 13 \]

7. **Calculate the distance**:
Now, substitute the values back into the distance formula:
\[ d = \frac{52}{13} = 4 \]

Thus, the length of the perpendicular from the point (3, -2) to the line \(12x - 5y + 6 = 0\) is **4**.