Find the distance of the point P from the lines AB in the following cases :
P(4, 2), AB is `5x-12y-9=0`

Answer

To find the distance of the point \( P(4, 2) \) from the line \( AB: 5x - 12y - 9 = 0 \), we will use the formula for the distance from a point to a line given by the equation \( Ax + By + C = 0 \).

### Step-by-step solution:

1. **Identify the coefficients from the line equation:**
The line equation is given as \( 5x - 12y - 9 = 0 \). We can rewrite this in the form \( Ax + By + C = 0 \) where:
- \( A = 5 \)
- \( B = -12 \)
- \( C = -9 \)

2. **Identify the coordinates of point P:**
The coordinates of point \( P \) are given as \( (x_1, y_1) = (4, 2) \).

3. **Substitute the values into the distance formula:**
The distance \( d \) from the point \( P(x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ d = \frac{|5(4) + (-12)(2) - 9|}{\sqrt{5^2 + (-12)^2}} \]

4. **Calculate the numerator:**
Calculate \( 5(4) + (-12)(2) - 9 \):
\[ = 20 - 24 - 9 = -13 \]
Taking the absolute value:
\[ | -13 | = 13 \]

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

6. **Calculate the distance:**
Now substitute back into the distance formula:
\[ d = \frac{13}{13} = 1 \]

### Final Answer:
The distance of the point \( P(4, 2) \) from the line \( AB: 5x - 12y - 9 = 0 \) is **1 unit**.