find the equation of the bisectors of the angle between 2 lines represented by the second degree homoegenous equation x^2-10xy+9y^2=0

The given equation is \( x^2 - 10xy + 9y^2 = 0 \), which represents two lines. To find the equation of the angle bisectors of these lines, we first express the given homogeneous quadratic equation as the product of two linear factors, and then apply the formula for the angle bisectors.

### Step 1: Factor the quadratic equation

We are given:
\[
x^2 - 10xy + 9y^2 = 0
\]
We need to factor this expression. To do so, we can treat it as a quadratic in \(x\) or \(y\), and find suitable factors.

Let's try factorizing it:
\[
x^2 - 10xy + 9y^2 = (x - y)(x - 9y) = 0
\]

Thus, the equation represents the pair of lines \( x - y = 0 \) and \( x - 9y = 0 \). These lines intersect at the origin.

### Step 2: Find the slopes of the lines

From the equations \( x - y = 0 \) and \( x - 9y = 0 \), we can deduce the slopes of the two lines:
- For \( x - y = 0 \), the slope is \( m_1 = 1 \).
- For \( x - 9y = 0 \), the slope is \( m_2 = \frac{1}{9} \).

### Step 3: Formula for the equation of the angle bisectors

The equation of the bisectors of the angle between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[
\frac{y}{x} = \frac{m_1 + m_2}{2} \quad \text{(for the bisector of the acute angle)}
\]
and
\[
\frac{y}{x} = \frac{m_1 - m_2}{2} \quad \text{(for the bisector of the obtuse angle)}
\]

### Step 4: Calculate the bisectors

Now, substitute \( m_1 = 1 \) and \( m_2 = \frac{1}{9} \) into the bisector formulas:

- Acute angle bisector:
\[
\frac{y}{x} = \frac{1 + \frac{1}{9}}{2} = \frac{\frac{10}{9}}{2} = \frac{5}{9}
\]
Thus, one bisector is \( y = \frac{5}{9}x \).

- Obtuse angle bisector:
\[
\frac{y}{x} = \frac{1 - \frac{1}{9}}{2} = \frac{\frac{8}{9}}{2} = \frac{4}{9}
\]
Thus, the other bisector is \( y = \frac{4}{9}x \).

### Final Answer:

The equations of the bisectors of the angle between the two lines are:
\[
y = \frac{5}{9}x \quad \text{(acute angle bisector)}
\]
and
\[
y = \frac{4}{9}x \quad \text{(obtuse angle bisector)}.
\]

Would you like further clarification or more details on any part of the solution?

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Here are 5 related questions to further expand your understanding:

1. How do you find the slopes of two lines from their general equation?
2. What are the conditions for two lines to form a pair of perpendicular bisectors?
3. Can the bisectors of two lines ever coincide with the lines themselves?
4. How would you find the angle between two lines given their slopes?
5. How can you determine if the angle between two lines is acute or obtuse?

**Tip:** The general formula for the bisector of two lines works for any angle, not just acute angles! Always check which angle (acute or obtuse) the formula gives you.

find the equation of the bisectors of the angle between 2 lines represented by the second degree homogeneous equation x^2 - 10xy + 9y^2 = 0

Learn how to find the equation of the bisectors of the angle between two lines from the homogeneous quadratic equation x^2 - 10xy + 9y^2 = 0. This step-by-step solution covers key concepts like factoring quadratic equations, calculating line slopes, and applying the angle bisector formulas. Suitable for Grades 11-12.

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