The equation `kx^2+4xy+5y^2=0` represents two lines inclined at an angle `pi` if `k` is

Options

A. `5//4`
B. `4//5`
C. `-45`
D. none of these

Explanation

To find the value of \( k \) for which the equation \( kx^2 + 4xy + 5y^2 = 0 \) represents two lines inclined at an angle \( \pi \), we can follow these steps:

### Step 1: Identify coefficients
The given equation is in the form of \( ax^2 + 2hxy + by^2 = 0 \). Here, we can identify:
- \( a = k \)
- \( 2h = 4 \) (thus \( h = 2 \))
- \( b = 5 \)

### Step 2: Use the angle condition
The angle \( \theta \) between the two lines represented by the equation is given by the formula:
\[ \tan \theta = \frac{\sqrt{h^2 - ab}}{a + b} \]
For \( \theta = \pi \), we know that \( \tan(\pi) = 0 \). Therefore, we set up the equation:
\[ \sqrt{h^2 - ab} = 0 \]

### Step 3: Substitute values
Substituting the values of \( h \), \( a \), and \( b \) into the equation:
\[ h^2 - ab = 0 \]
\[ 2^2 - (k)(5) = 0 \]
This simplifies to:
\[ 4 - 5k = 0 \]

### Step 4: Solve for \( k \)
Rearranging the equation gives:
\[ 5k = 4 \]
\[ k = \frac{4}{5} \]

### Conclusion
The value of \( k \) for which the equation \( kx^2 + 4xy + 5y^2 = 0 \) represents two lines inclined at an angle \( \pi \) is:
\[ \boxed{\frac{4}{5}} \]