The equations of a line which is parallel to the line common to the pair of lines given by `6x^(2)-xy-12y^(2)=0and15x^(2)+14xy-8y^(2)=0`and at a distance of 7 units from it is :

Options

A. `3x-4y=-35`
B. `5x-2y=7`
C. `3x+4y=35`
D. `2x-3y=7`

Explanation

To find the equations of the lines that are parallel to the common line of the given pair of lines and at a distance of 7 units from it, we can follow these steps:

### Step 1: Identify the equations of the given pair of lines
The given equations are:
1. \( 6x^2 - xy - 12y^2 = 0 \)
2. \( 15x^2 + 14xy - 8y^2 = 0 \)

### Step 2: Factor the first equation
We can factor the first equation as follows:
\[ 6x^2 - xy - 12y^2 = 0 \]
Rearranging gives:
\[ 6x^2 - 9xy + 8xy - 12y^2 = 0 \]
Factoring by grouping:
\[ 3x(2x - 3y) + 4y(2x - 3y) = 0 \]
This can be factored as:
\[ (3x + 4y)(2x - 3y) = 0 \]
Thus, the lines are:
1. \( 3x + 4y = 0 \)
2. \( 2x - 3y = 0 \)

### Step 3: Factor the second equation
Now, we factor the second equation:
\[ 15x^2 + 14xy - 8y^2 = 0 \]
Rearranging gives:
\[ 15x^2 + 20xy - 6xy - 8y^2 = 0 \]
Factoring by grouping:
\[ 5x(3x + 4y) - 2y(3x + 4y) = 0 \]
This can be factored as:
\[ (5x - 2y)(3x + 4y) = 0 \]
Thus, the lines are:
1. \( 5x - 2y = 0 \)
2. \( 3x + 4y = 0 \)

### Step 4: Identify the common line
The common line from the factored forms is:
\[ 3x + 4y = 0 \]

### Step 5: Find the equation of the parallel line
We need to find a line parallel to \( 3x + 4y = 0 \) that is at a distance of 7 units from it. The general form of a line parallel to \( 3x + 4y = 0 \) can be written as:
\[ 3x + 4y + k = 0 \]

### Step 6: Use the distance formula
The distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 3 \), \( B = 4 \), \( C_1 = 0 \) (from \( 3x + 4y = 0 \)), and \( C_2 = k \). Setting \( d = 7 \):
\[ 7 = \frac{|0 - k|}{\sqrt{3^2 + 4^2}} = \frac{|k|}{\sqrt{9 + 16}} = \frac{|k|}{5} \]

### Step 7: Solve for \( k \)
From the equation:
\[ 7 = \frac{|k|}{5} \]
Multiplying both sides by 5 gives:
\[ |k| = 35 \]
Thus, \( k = 35 \) or \( k = -35 \).

### Step 8: Write the equations of the parallel lines
The equations of the lines parallel to \( 3x + 4y = 0 \) and at a distance of 7 units are:
1. \( 3x + 4y + 35 = 0 \)
2. \( 3x + 4y - 35 = 0 \)

### Final Answer
The equations of the lines are:
1. \( 3x + 4y + 35 = 0 \)
2. \( 3x + 4y - 35 = 0 \)

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