The equation `x-y=4 and x^2+4xy+y^2=0` represent the sides of

Options

A. an equilateral triangle
B. a right - angled triangle
C. an isosceles triangle
D. None of these

Explanation

To determine the type of triangle represented by the equations \( x - y = 4 \) and \( x^2 + 4xy + y^2 = 0 \), we will follow these steps:

### Step 1: Identify the equations
The first equation is a linear equation representing a line:
\[ x - y = 4 \quad \text{(Equation 1)} \]
The second equation is a quadratic equation:
\[ x^2 + 4xy + y^2 = 0 \quad \text{(Equation 2)} \]

### Step 2: Rewrite Equation 2 in standard form
The standard form of a conic section is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
For our equation \( x^2 + 4xy + y^2 = 0 \), we can identify:
- \( A = 1 \)
- \( B = 4 \)
- \( C = 1 \)

### Step 3: Calculate the angle between the lines
The angle \( \theta \) between two lines represented by the equations \( Ax^2 + Bxy + Cy^2 = 0 \) can be calculated using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
where \( a = A \), \( b = C \), and \( h = \frac{B}{2} \).

Substituting the values:
- \( a = 1 \)
- \( b = 1 \)
- \( h = \frac{4}{2} = 2 \)

We find:
\[ \tan \theta = \frac{2 \sqrt{2^2 - (1)(1)}}{1 + 1} = \frac{2 \sqrt{4 - 1}}{2} = \frac{2 \sqrt{3}}{2} = \sqrt{3} \]
Thus,
\[ \theta = \tan^{-1}(\sqrt{3}) = 60^\circ \]

### Step 4: Determine the type of triangle
Since the angle between the two lines is \( 60^\circ \), we can analyze the triangle formed by these lines. The line \( x - y = 4 \) is one side of the triangle, and the other side is determined by the quadratic equation.

### Step 5: Check for perpendicularity
The angle bisector of the angle between the two lines can be found. The angle bisector equations are derived from the coefficients of the quadratic equation. In this case, the angle bisector is perpendicular to the line \( x - y = 4 \).

Since the angle between the two lines is \( 60^\circ \), and the angle bisector creates two \( 30^\circ \) angles, we conclude that the triangle formed is isosceles.

### Conclusion
Since we have two equal angles (each \( 30^\circ \)) and one angle of \( 60^\circ \), this triangle is also equilateral.

Thus, the equations \( x - y = 4 \) and \( x^2 + 4xy + y^2 = 0 \) represent the sides of an **equilateral triangle**.