The equation of the lines represented by `x^(2)+2(cosecalpha)xy+y^(2)=0` are

Options

A. `(1pmsinalpha)x+2(cosalpha)y=0`
B. `(1pmcosalpha)x+2(sinalpha)y=0`
C. `(1pmsinalpha)x+(cosalpha)y=0`
D. `(1pmcosalpha)x+(sinalpha)y=0`

Answer

To solve the problem of finding the equations of the lines represented by the equation \(x^2 + 2(\cos \alpha)xy + y^2 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^2 + 2(\cos \alpha)xy + y^2 = 0 \]

### Step 2: Divide by \(x^2\)
To simplify the equation, we divide the entire equation by \(x^2\):
\[ 1 + 2(\cos \alpha)\frac{y}{x} + \left(\frac{y}{x}\right)^2 = 0 \]
Let \(m = \frac{y}{x}\), then the equation becomes:
\[ 1 + 2(\cos \alpha)m + m^2 = 0 \]

### Step 3: Rearrange into standard quadratic form
Rearranging gives us a standard quadratic equation in \(m\):
\[ m^2 + 2(\cos \alpha)m + 1 = 0 \]

### Step 4: Apply the quadratic formula
We can now use the quadratic formula to solve for \(m\):
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 2(\cos \alpha)\), and \(c = 1\). Plugging these values in:
\[ m = \frac{-2(\cos \alpha) \pm \sqrt{(2(\cos \alpha))^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ m = \frac{-2(\cos \alpha) \pm \sqrt{4\cos^2 \alpha - 4}}{2} \]
\[ m = \frac{-2(\cos \alpha) \pm 2\sqrt{\cos^2 \alpha - 1}}{2} \]
\[ m = -\cos \alpha \pm \sqrt{\cos^2 \alpha - 1} \]

### Step 5: Simplify the square root
Since \(\cos^2 \alpha - 1 = -\sin^2 \alpha\), we have:
\[ m = -\cos \alpha \pm i\sin \alpha \]
This gives us:
\[ m = -\cos \alpha \pm i\sin \alpha \]

### Step 6: Write the equations of the lines
The slopes of the lines can be expressed as:
\[ m_1 = -\cos \alpha + i\sin \alpha \quad \text{and} \quad m_2 = -\cos \alpha - i\sin \alpha \]
Thus, the equations of the lines can be written as:
\[ y = m_1 x \quad \text{and} \quad y = m_2 x \]
This leads to:
\[ y + \cos \alpha x - i\sin \alpha x = 0 \quad \text{and} \quad y + \cos \alpha x + i\sin \alpha x = 0 \]

### Final Result
The equations of the lines represented by the given equation are:
\[ y + \cos \alpha x - i\sin \alpha x = 0 \quad \text{and} \quad y + \cos \alpha x + i\sin \alpha x = 0 \]