If the angle between the lines given by `6x^(2)+xy+ky^(2)=0` is `45^(@)`, then k=

Options

A. `1, 35`
B. `-1, 35`
C. `1,-35`
D. `-1, -35`

Answer

To solve the problem, we need to find the value of \( k \) such that the angle between the lines represented by the equation \( 6x^2 + xy + ky^2 = 0 \) is \( 45^\circ \).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is in the form \( ax^2 + bxy + cy^2 = 0 \). Here, we can identify:
- \( a = 6 \)
- \( b = 1 \) (since \( xy \) has a coefficient of 1)
- \( c = k \)

2. **Use the formula for the angle between two lines**:
The angle \( \theta \) between the two lines represented by the equation can be calculated using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
where \( h = \frac{b}{2} \).

3. **Substitute the values**:
Here, \( h = \frac{1}{2} \). Now we can substitute \( a \), \( b \), and \( c \) into the formula:
\[ \tan 45^\circ = \frac{2\sqrt{\left(\frac{1}{2}\right)^2 - (6)(k)}}{6 + k} \]
Since \( \tan 45^\circ = 1 \), we have:
\[ 1 = \frac{2\sqrt{\frac{1}{4} - 6k}}{6 + k} \]

4. **Cross-multiply**:
Cross-multiplying gives:
\[ 6 + k = 2\sqrt{\frac{1}{4} - 6k} \]

5. **Square both sides**:
Squaring both sides to eliminate the square root:
\[ (6 + k)^2 = 4\left(\frac{1}{4} - 6k\right) \]

6. **Expand and simplify**:
Expanding both sides:
\[ 36 + 12k + k^2 = 1 - 24k \]
Rearranging gives:
\[ k^2 + 12k + 24k + 36 - 1 = 0 \]
Simplifying further:
\[ k^2 + 36k + 35 = 0 \]

7. **Factor the quadratic equation**:
We can factor this as:
\[ (k + 1)(k + 35) = 0 \]

8. **Find the values of \( k \)**:
Setting each factor to zero gives us:
\[ k + 1 = 0 \quad \Rightarrow \quad k = -1 \]
\[ k + 35 = 0 \quad \Rightarrow \quad k = -35 \]

Thus, the values of \( k \) are \( k = -1 \) or \( k = -35 \).

### Final Answer:
The values of \( k \) are \( -1 \) and \( -35 \).