The three lines whose combined equation is `y^(3)-4x^(2)y=0` form a triangle which is

Options

A. isosceles
B. equilateral
C. right angled
D. none of these

Explanation

To solve the problem, we need to analyze the combined equation of the three lines given by \( y^3 - 4x^2y = 0 \) and determine the type of triangle they form.

### Step-by-Step Solution:

1. **Factor the Combined Equation**:
The first step is to factor the given equation \( y^3 - 4x^2y = 0 \).
\[ y(y^2 - 4x^2) = 0 \]
This gives us two parts: \( y = 0 \) and \( y^2 - 4x^2 = 0 \).

2. **Solve for the Lines**:
From the factored form, we can derive the lines:
- From \( y = 0 \), we have the first line: \( L_1: y = 0 \).
- From \( y^2 - 4x^2 = 0 \), we can rewrite it as \( (y - 2x)(y + 2x) = 0 \), leading to:
- Second line: \( L_2: y = 2x \)
- Third line: \( L_3: y = -2x \)

3. **Identify the Lines**:
We have identified the three lines:
- \( L_1: y = 0 \) (the x-axis)
- \( L_2: y = 2x \) (a line with a positive slope)
- \( L_3: y = -2x \) (a line with a negative slope)

4. **Graph the Lines**:
All three lines intersect at the origin (0,0). The lines \( y = 2x \) and \( y = -2x \) are symmetric about the x-axis and will form angles with the x-axis.

5. **Determine the Type of Triangle**:
Since all three lines pass through the origin and do not enclose any area, they do not form a triangle. Instead, they intersect at a single point (the origin) and extend infinitely in both directions.

6. **Conclusion**:
Therefore, the three lines do not form any triangle structure. The answer is that they do not form any type of triangle.