The equation of line which is parallel to the line common to the pair of lines given by `3x^2+xy-4y^2=0 and 6x^2+11xy+4y^2=0` and at a distance of 2 units from it is

Options

A. `3x-4y=-10`
B. `x-y=2`
C. `3x+4y=10`
D. `2x+y=-2`

Answer

To find the equation of the line that is parallel to the line common to the pair of lines given by \(3x^2 + xy - 4y^2 = 0\) and \(6x^2 + 11xy + 4y^2 = 0\) and is at a distance of 2 units from it, we can follow these steps:

### Step 1: Factor the first equation
The first equation is \(3x^2 + xy - 4y^2 = 0\). We can factor this as follows:

\[ 3x^2 + xy - 4y^2 = 0 \implies (3x - 4y)(x + y) = 0 \]

This gives us two lines: \(3x - 4y = 0\) and \(x + y = 0\).

### Step 2: Factor the second equation
Now, let's factor the second equation \(6x^2 + 11xy + 4y^2 = 0\):

\[ 6x^2 + 11xy + 4y^2 = 0 \implies (2x + y)(3x + 4y) = 0 \]

This gives us two lines: \(2x + y = 0\) and \(3x + 4y = 0\).

### Step 3: Identify the common line
From the factored forms, the common line between the two pairs of lines is \(3x + 4y = 0\).

### Step 4: Write the equation of the parallel line
Since we need a line parallel to \(3x + 4y = 0\), we can express it in the form:

\[ 3x + 4y + k = 0 \]

where \(k\) is a constant.

### Step 5: 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:

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

In our case, \(A = 3\), \(B = 4\), \(C_1 = 0\) (from the line \(3x + 4y = 0\)), and \(C_2 = k\). We know that the distance \(d\) is 2 units:

\[ 2 = \frac{|k - 0|}{\sqrt{3^2 + 4^2}} = \frac{|k|}{5} \]

### Step 6: Solve for \(k\)
Now, we can solve for \(k\):

\[ 2 = \frac{|k|}{5} \implies |k| = 10 \]

Thus, \(k = 10\) or \(k = -10\).

### Step 7: Write the final equations
Substituting back, we have two parallel lines:

1. \(3x + 4y + 10 = 0\)
2. \(3x + 4y - 10 = 0\)

### Final Answer
The equations of the lines that are parallel to the common line and at a distance of 2 units from it are:

\[ 3x + 4y + 10 = 0 \quad \text{and} \quad 3x + 4y - 10 = 0 \]