The equation to the pair of lines perpendicular to the pair of lines `3x^(2)-4xy+y^(2)=0`, is

Options

A. `x^(2)+4xy+3y^(2)=0`
B. `x^(2)-4xy-3y^(2)=0`
C. `x^(2)+4xy+y^(2)=0`
D. none of these

Explanation

To find the equation of the pair of lines that are perpendicular to the given pair of lines represented by the equation \(3x^2 - 4xy + y^2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is in the form \(Ax^2 + Bxy + Cy^2 = 0\), where:
- \(A = 3\)
- \(B = -4\)
- \(C = 1\)

### Step 2: Change the coefficients
To find the equation of the pair of lines that are perpendicular to the original lines, we will change the coefficients according to the following rules:
- The coefficient of \(x^2\) becomes the coefficient of \(y^2\).
- The coefficient of \(y^2\) becomes the coefficient of \(x^2\).
- The sign of the coefficient of \(xy\) changes.

Thus, we will have:
- New coefficient of \(x^2\) = \(C = 1\)
- New coefficient of \(y^2\) = \(A = 3\)
- New coefficient of \(xy\) = \(-B = 4\)

### Step 3: Write the new equation
Using the new coefficients, we can write the equation of the pair of lines that are perpendicular:
\[ 1x^2 + 4xy + 3y^2 = 0 \]
or simply,
\[ x^2 + 4xy + 3y^2 = 0 \]

### Step 4: Conclusion
Thus, the equation of the pair of lines that are perpendicular to the given pair of lines is:
\[ x^2 + 4xy + 3y^2 = 0 \]