Find the equation of the bisectors of the angle between the lines represented by `3x^2-5xy+4y^2=0`

Answer

To find the equation of the bisectors of the angle between the lines represented by the equation \(3x^2 - 5xy + 4y^2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is in the form \(Ax^2 + Bxy + Cy^2 = 0\), where:
- \(A = 3\)
- \(B = -5\)
- \(C = 4\)

### Step 2: Use the formula for angle bisectors
The formula for the angle bisectors of the lines represented by the equation \(Ax^2 + Bxy + Cy^2 = 0\) is given by:
\[ H x^2 - (A + C) y^2 - B xy = 0 \]
where \(H = \frac{B}{2}\).

### Step 3: Calculate H
Calculate \(H\):
\[ H = \frac{-5}{2} = -\frac{5}{2} \]

### Step 4: Substitute values into the angle bisector formula
Now substitute \(H\), \(A\), \(B\), and \(C\) into the angle bisector formula:
\[ -\frac{5}{2} x^2 - (3 + 4) y^2 - (-5) xy = 0 \]

### Step 5: Simplify the equation
This simplifies to:
\[ -\frac{5}{2} x^2 - 7y^2 + 5xy = 0 \]
To eliminate the fraction, multiply the entire equation by \(-2\):
\[ 5x^2 + 14y^2 - 10xy = 0 \]

### Step 6: Rearrange the equation
Rearranging gives us the final equation of the angle bisector:
\[ 5x^2 - 10xy + 14y^2 = 0 \]

### Final Answer
The equation of the bisectors of the angle between the lines represented by \(3x^2 - 5xy + 4y^2 = 0\) is:
\[ 5x^2 - 10xy + 14y^2 = 0 \]

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