The perpendicular distance between the straight lines 6x + 8y + 15 =0 and 3x + 4y + 9 =0 is

Options

A. 3/2 units
B. 3/10 units
C. 3/4 units
D. 2/7 units

Answer

To find the perpendicular distance between the two straight lines given by the equations \(6x + 8y + 15 = 0\) and \(3x + 4y + 9 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the two lines are:
1. Line 1: \(6x + 8y + 15 = 0\)
2. Line 2: \(3x + 4y + 9 = 0\)

### Step 2: Rewrite the second line in the same form as the first line
To use the formula for the distance between two parallel lines, we need to ensure that the coefficients of \(x\) and \(y\) are the same in both equations. We can do this by multiplying the second line by 2:

\[ 2(3x + 4y + 9) = 0 \implies 6x + 8y + 18 = 0 \]

Now we have:
1. Line 1: \(6x + 8y + 15 = 0\)
2. Line 2: \(6x + 8y + 18 = 0\)

### Step 3: Identify coefficients for the distance formula
The general form of the equation of a line is \(Ax + By + C = 0\). Here, we have:
- For Line 1: \(A = 6\), \(B = 8\), \(C_1 = 15\)
- For Line 2: \(A = 6\), \(B = 8\), \(C_2 = 18\)

### Step 4: Use the formula for the distance between two parallel lines
The formula for the perpendicular distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

Substituting the values we have:

\[ d = \frac{|18 - 15|}{\sqrt{6^2 + 8^2}} \]

### Step 5: Calculate the distance
Calculating the numerator:

\[ |C_2 - C_1| = |18 - 15| = 3 \]

Calculating the denominator:

\[ \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Now substituting back into the formula:

\[ d = \frac{3}{10} \]

### Final Answer
The perpendicular distance between the two lines is:

\[ \frac{3}{10} \]

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