The equation `3x^(2)-8xy-3y^(2)=0` and `x+2y=3` represent the sides of a triangle which is

Options

A. equilateral
B. right angled
C. isosceles
D. isosceles rt. Angled

Answer

To solve the problem, we need to analyze the given equations and determine the nature of the triangle formed by the lines represented by these equations.

### Step-by-Step Solution:

1. **Identify the given equations**:
We have two equations:
- Equation 1: \(3x^2 - 8xy - 3y^2 = 0\)
- Equation 2: \(x + 2y = 3\)

2. **Analyze the first equation**:
The first equation represents a pair of straight lines. To confirm this, we can check the condition for a pair of straight lines in the form \(Ax^2 + Bxy + Cy^2 = 0\).
Here, \(A = 3\), \(B = -8\), and \(C = -3\).

3. **Check the condition for perpendicular lines**:
For the lines to be perpendicular, the condition is:
\[ A + C = 0 \]
Substituting the values:
\[ 3 + (-3) = 0 \]
This condition is satisfied, indicating that the lines represented by the first equation are perpendicular.

4. **Analyze the second equation**:
The second equation \(x + 2y = 3\) can be rewritten in slope-intercept form:
\[ y = -\frac{1}{2}x + \frac{3}{2} \]
The slope of this line is \(-\frac{1}{2}\).

5. **Find the slopes of the lines from the first equation**:
To find the slopes of the lines represented by the first equation, we can factor the quadratic equation or use the formula for the slopes of the lines given by:
\[ m_1, m_2 = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values:
\[ m_1, m_2 = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3} \]
\[ = \frac{8 \pm \sqrt{64 + 36}}{6} = \frac{8 \pm \sqrt{100}}{6} = \frac{8 \pm 10}{6} \]
This gives us:
\[ m_1 = \frac{18}{6} = 3, \quad m_2 = \frac{-2}{6} = -\frac{1}{3} \]

6. **Check for perpendicularity**:
The product of the slopes of two lines is \(-1\) for them to be perpendicular:
\[ m_1 \cdot m_2 = 3 \cdot \left(-\frac{1}{3}\right) = -1 \]
This confirms that the lines from the first equation are indeed perpendicular.

7. **Conclusion**:
Since we have established that the lines from the first equation are perpendicular and the second equation represents a line, the triangle formed by these lines is a right-angled triangle.

### Final Answer:
The triangle formed by the equations \(3x^2 - 8xy - 3y^2 = 0\) and \(x + 2y = 3\) is a right-angled triangle.