Find the equations of the line through the intersection of `2x - 3y + 4 = 0 and 3x + 4y - 5= 0 `and perpendicular to `6x-7y +c = 0 `

Options

A. `119 y + 20 x = 125`
B. `199 y - 120 x = 125`
C. `119x + 102 y = 125`
D. `119 x - 102 y = 125`

Answer

To find the equations of the line through the intersection of the lines \(2x - 3y + 4 = 0\) and \(3x + 4y - 5 = 0\) that is also perpendicular to the line \(6x - 7y + c = 0\), we can follow these steps:

### Step 1: Find the intersection of the two lines
We have the equations:
1. \(2x - 3y + 4 = 0\)
2. \(3x + 4y - 5 = 0\)

We can solve these equations simultaneously.

**Rearranging the first equation:**
\[ 2x - 3y = -4 \quad \text{(1)} \]

**Rearranging the second equation:**
\[ 3x + 4y = 5 \quad \text{(2)} \]

Now, we can solve for \(x\) and \(y\). We can multiply equation (1) by 3 and equation (2) by 2 to eliminate \(x\):
\[ 6x - 9y = -12 \quad \text{(3)} \]
\[ 6x + 8y = 10 \quad \text{(4)} \]

Now, subtract equation (4) from equation (3):
\[ (6x - 9y) - (6x + 8y) = -12 - 10 \]
\[ -17y = -22 \]
\[ y = \frac{22}{17} \]

Now substitute \(y\) back into equation (1) to find \(x\):
\[ 2x - 3\left(\frac{22}{17}\right) + 4 = 0 \]
\[ 2x - \frac{66}{17} + \frac{68}{17} = 0 \]
\[ 2x + \frac{2}{17} = 0 \]
\[ 2x = -\frac{2}{17} \]
\[ x = -\frac{1}{17} \]

So, the intersection point is \(\left(-\frac{1}{17}, \frac{22}{17}\right)\).

### Step 2: Find the slope of the line \(6x - 7y + c = 0\)
The slope \(m_1\) of the line \(6x - 7y + c = 0\) can be calculated as follows:
\[ m_1 = -\frac{A}{B} = -\frac{6}{-7} = \frac{6}{7} \]

### Step 3: Find the slope of the line perpendicular to it
The slope \(m_2\) of the line we need to find, which is perpendicular to the line \(6x - 7y + c = 0\), is given by:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{6}{7}} = -\frac{7}{6} \]

### Step 4: Use point-slope form to find the equation of the line
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\frac{7}{6}\), \(x_1 = -\frac{1}{17}\), and \(y_1 = \frac{22}{17}\):
\[ y - \frac{22}{17} = -\frac{7}{6}\left(x + \frac{1}{17}\right) \]

### Step 5: Simplify the equation
Multiply both sides by 102 (the least common multiple of 6 and 17) to eliminate the fractions:
\[ 102y - 132 = -119x - 7 \]
Rearranging gives:
\[ 119x + 102y = 125 \]

### Final Answer
The equation of the line through the intersection of the two lines and perpendicular to \(6x - 7y + c = 0\) is:
\[ 119x + 102y = 125 \]

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