The distance of the point (1,-1) from the line 12(x-6) = 5( y+2) is

Options

A. (a) 5 units
B. (b) 25 units
C. (c) `(5)/(2)` units
D. (d) `(25)/(4)` units

Answer

To find the distance of the point (1, -1) from the line given by the equation \(12(x - 6) = 5(y + 2)\), we will follow these steps:

### Step 1: Rearranging the Line Equation
First, we need to rearrange the line equation into the standard form \(Ax + By + C = 0\).

Starting with the given equation:
\[ 12(x - 6) = 5(y + 2) \]

Expanding both sides:
\[ 12x - 72 = 5y + 10 \]

Rearranging gives:
\[ 12x - 5y - 72 - 10 = 0 \]
\[ 12x - 5y - 82 = 0 \]

### Step 2: Identifying Coefficients
From the equation \(12x - 5y - 82 = 0\), we can identify:
- \(A = 12\)
- \(B = -5\)
- \(C = -82\)

### Step 3: Using the Distance Formula
The formula for the distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Substituting the point \((1, -1)\) into the formula:
- \(x_1 = 1\)
- \(y_1 = -1\)

### Step 4: Calculating the Numerator
Calculating \(Ax_1 + By_1 + C\):
\[ d = \frac{|12(1) + (-5)(-1) - 82|}{\sqrt{12^2 + (-5)^2}} \]

Calculating inside the absolute value:
\[ = |12 + 5 - 82| = |17 - 82| = |-65| = 65 \]

### Step 5: Calculating the Denominator
Calculating \(\sqrt{A^2 + B^2}\):
\[ \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 6: Final Calculation of Distance
Now substituting back into the distance formula:
\[ d = \frac{65}{13} = 5 \]

### Conclusion
The distance from the point (1, -1) to the line is \(5\) units.