The equation of the lines parallel to the line common to the pair of lines given by `6x^(2)-xy-12y^(2)=0` and `15x^(2)+14xy-8y^(2)=0` and the sum of whose intercepts on the axes is 7, is

Options

A. `2x-3y=42`
B. `3x+4y=12`
C. `5x-2y=10`
D. none of these

Answer

To solve the problem, we need to find the equation of the lines that are parallel to the common line of the given pairs of lines and also satisfy the condition that the sum of their intercepts on the axes is 7.

### Step 1: Find the common line from the given pairs of lines.

The first pair of lines is given by the equation:
\[ 6x^2 - xy - 12y^2 = 0 \]

To factor this, we can rewrite it as:
\[ (3x + 4y)(2x - 3y) = 0 \]

This gives us two lines:
1. \( 3x + 4y = 0 \)
2. \( 2x - 3y = 0 \)

The second pair of lines is given by the equation:
\[ 15x^2 + 14xy - 8y^2 = 0 \]

Factoring this, we can rewrite it as:
\[ (3x + 4y)(5x - 2y) = 0 \]

This gives us two lines:
1. \( 3x + 4y = 0 \)
2. \( 5x - 2y = 0 \)

From both pairs, we see that the common line is:
\[ 3x + 4y = 0 \]

### Step 2: Find the equation of the lines parallel to the common line.

The general form of the line parallel to \( 3x + 4y = 0 \) can be written as:
\[ 3x + 4y = c \]
where \( c \) is a constant.

### Step 3: Convert the equation to intercept form.

To find the intercepts, we convert the equation \( 3x + 4y = c \) into intercept form:
\[ \frac{x}{\frac{c}{3}} + \frac{y}{\frac{c}{4}} = 1 \]

Here, the x-intercept \( a = \frac{c}{3} \) and the y-intercept \( b = \frac{c}{4} \).

### Step 4: Use the condition on intercepts.

We know that the sum of the intercepts is given as:
\[ a + b = 7 \]

Substituting the intercepts:
\[ \frac{c}{3} + \frac{c}{4} = 7 \]

### Step 5: Solve for \( c \).

To solve for \( c \), we need a common denominator:
\[ \frac{4c + 3c}{12} = 7 \]
\[ \frac{7c}{12} = 7 \]

Multiplying both sides by 12:
\[ 7c = 84 \]
\[ c = 12 \]

### Step 6: Write the final equation.

Now substituting \( c \) back into the equation of the line:
\[ 3x + 4y = 12 \]

This is the required equation of the lines parallel to the common line and whose intercepts sum to 7.

### Final Answer:
The equation of the lines is:
\[ 3x + 4y = 12 \]

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