Find the perpendicular distance between the lines.
`3x+4y-5=0 ...(1) and 6x+8y-45=0 ...(2)`

Answer

To find the perpendicular distance between the two lines given by the equations \(3x + 4y - 5 = 0\) and \(6x + 8y - 45 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The two lines are:
1. \(3x + 4y - 5 = 0\) (Equation 1)
2. \(6x + 8y - 45 = 0\) (Equation 2)

### Step 2: Simplify the second equation
We can simplify the second equation by dividing all terms by 2:
\[ 6x + 8y - 45 = 0 \implies 3x + 4y - \frac{45}{2} = 0 \]
Now, we can rewrite the second equation as:
\[ 3x + 4y - \frac{45}{2} = 0 \quad (Equation 2') \]

### Step 3: Confirm that the lines are parallel
Both equations can be expressed in the form \(Ax + By + C = 0\):
- For Equation 1: \(A = 3\), \(B = 4\), \(C_1 = -5\)
- For Equation 2': \(A = 3\), \(B = 4\), \(C_2 = -\frac{45}{2}\)

Since the coefficients of \(x\) and \(y\) are the same in both equations, the lines are parallel.

### Step 4: Use the formula for the distance between two parallel lines
The formula for the distance \(d\) between two parallel lines given by the equations \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step 5: Substitute the values into the formula
Here, we have:
- \(C_1 = -5\)
- \(C_2 = -\frac{45}{2}\)
- \(A = 3\)
- \(B = 4\)

Now, substituting these values into the formula:
\[ d = \frac{\left| -\frac{45}{2} - (-5) \right|}{\sqrt{3^2 + 4^2}} \]

### Step 6: Simplify the numerator
Calculating the numerator:
\[ -\frac{45}{2} + 5 = -\frac{45}{2} + \frac{10}{2} = -\frac{35}{2} \]
Thus, the absolute value is:
\[ \left| -\frac{35}{2} \right| = \frac{35}{2} \]

### Step 7: Calculate the denominator
Now, calculate the denominator:
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 8: Final calculation of the distance
Now we can find \(d\):
\[ d = \frac{\frac{35}{2}}{5} = \frac{35}{2 \times 5} = \frac{35}{10} = 3.5 \]

### Conclusion
The perpendicular distance between the two lines is \(3.5\) units.

---