The lines given by the equation `(2y^2+3xy-2x^2)(x+y-1)=0` form a triangle which is

Options

A. equilateral
B. isosceles
C. right angled
D. obtuse angled

Answer

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

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation provided is \((2y^2 + 3xy - 2x^2)(x + y - 1) = 0\). This indicates that we have two parts: the quadratic part \(2y^2 + 3xy - 2x^2 = 0\) and the linear part \(x + y - 1 = 0\).

2. **Analyze the quadratic part**:
The quadratic equation \(2y^2 + 3xy - 2x^2 = 0\) can be treated as a pair of straight lines. To find the slopes of these lines, we can rewrite this equation in the standard form of a conic section.

3. **Using the condition for a pair of straight lines**:
The general form for a pair of straight lines is \(Ax^2 + Bxy + Cy^2 = 0\). Here, \(A = -2\), \(B = 3\), and \(C = 2\). The condition for this to represent a pair of straight lines is that \(B^2 - 4AC = 0\).

\[ B^2 - 4AC = 3^2 - 4(-2)(2) = 9 + 16 = 25 \quad (\text{not equal to } 0) \]

Since \(B^2 - 4AC > 0\), this confirms that we indeed have a pair of straight lines.

4. **Finding the slopes of the lines**:
The slopes of the lines can be found using the formula for the roots of the quadratic equation. The slopes \(m_1\) and \(m_2\) can be calculated using the quadratic formula:

\[ m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting the values:

\[ m = \frac{-3 \pm \sqrt{25}}{2 \cdot 2} = \frac{-3 \pm 5}{4} \]

This gives us two slopes:

\[ m_1 = \frac{2}{4} = \frac{1}{2}, \quad m_2 = \frac{-8}{4} = -2 \]

5. **Check for perpendicularity**:
To check if the lines are perpendicular, we can use the condition that the product of their slopes should equal -1:

\[ m_1 \cdot m_2 = \frac{1}{2} \cdot (-2) = -1 \]

Since the product of the slopes is -1, the lines are perpendicular.

6. **Analyze the linear part**:
The linear equation \(x + y - 1 = 0\) represents a straight line.

7. **Conclusion about the triangle**:
Since we have two lines that are perpendicular to each other and intersect with a third line, they form a right triangle. Therefore, the triangle formed by these lines is a right-angled triangle.

### Final Answer:
The triangle formed by the lines is a right-angled triangle.