Find the equation of the bisectors of the angles between the lines joining the origin to the point of intersection of the straight line `x-y=2` with the curve `5x^2+11 x y+8y^2+8x-4y+12=0`

Explanation

To find the equation of the bisectors of the angles between the lines joining the origin to the point of intersection of the straight line \( x - y = 2 \) with the curve \( 5x^2 + 11xy + 8y^2 + 8x - 4y + 12 = 0 \), we can follow these steps:

### Step 1: Find the intersection points of the line and the curve.
We start by substituting \( y = x - 2 \) (from the line equation \( x - y = 2 \)) into the curve equation.

**Substitution:**
\[ 5x^2 + 11x(x - 2) + 8(x - 2)^2 + 8x - 4(x - 2) + 12 = 0 \]

### Step 2: Simplify the equation.
Expanding the equation:
\[ 5x^2 + 11x^2 - 22x + 8(x^2 - 4x + 4) + 8x - 4x + 8 + 12 = 0 \]
\[ 5x^2 + 11x^2 + 8x^2 - 22x - 32x + 40 + 8 + 12 = 0 \]
Combine like terms:
\[ (5 + 11 + 8)x^2 + (-22 - 32)x + (40 + 8 + 12) = 0 \]
\[ 24x^2 - 54x + 60 = 0 \]

### Step 3: Solve the quadratic equation.
We can simplify this equation by dividing through by 6:
\[ 4x^2 - 9x + 10 = 0 \]
Now we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot 10}}{2 \cdot 4} \]
\[ x = \frac{9 \pm \sqrt{81 - 160}}{8} \]
\[ x = \frac{9 \pm \sqrt{-79}}{8} \]
Since the discriminant is negative, there are no real intersection points.

### Step 4: Find the angle bisectors.
Since we have no real intersection points, we can still find the angle bisectors of the lines joining the origin to the points that would have been the intersection points if they existed.

The general form of the angle bisector of two lines \( L_1: y = m_1x \) and \( L_2: y = m_2x \) is given by:
\[ \frac{y}{x} = \frac{m_1 + m_2 \pm \sqrt{(m_1^2 + 1)(m_2^2 + 1)}}{1 - m_1m_2} \]

### Step 5: Write the final equation of the bisectors.
In this case, since we do not have specific slopes from intersection points, we can express the angle bisector in terms of the slopes of the lines formed by the intersection points (if they existed).

The equation of the angle bisectors can be represented as:
\[ x^2 - y^2 = kxy \]
where \( k \) is a constant derived from the slopes of the lines.

### Final Result:
The equation of the bisectors of the angles between the lines joining the origin to the intersection points (if they existed) can be expressed in the form:
\[ x^2 - y^2 = kxy \]