If one of the lines given by `kx^(2)+xy-y^(2)=0` bisects the angle between the co-ordinate axes, then k=

Options

A. `1, -1`
B. `0, 1`
C. `0, -2`
D. `0, 2`

Answer

To solve the problem, we need to find the value of \( k \) such that one of the lines given by the equation \( kx^2 + xy - y^2 = 0 \) bisects the angle between the coordinate axes.

### Step 1: Understand the Equation
The given equation is \( kx^2 + xy - y^2 = 0 \). This is a quadratic equation in terms of \( x \) and \( y \).

### Step 2: Rewrite the Equation
We can rewrite the equation in a more manageable form. Dividing the entire equation by \( x^2 \) (assuming \( x \neq 0 \)):
\[ k + \frac{y}{x} - \left(\frac{y}{x}\right)^2 = 0 \]
Let \( m = \frac{y}{x} \) (the slope of the line). Then the equation becomes:
\[ -k + m - m^2 = 0 \]
or rearranging gives:
\[ m^2 - m - k = 0 \]

### Step 3: Find the Slopes
The lines that bisect the angle between the coordinate axes have slopes of \( m_1 = 1 \) (for \( 45^\circ \)) and \( m_2 = -1 \) (for \( 135^\circ \)).

### Step 4: Substitute the Slopes
We will substitute \( m = 1 \) and \( m = -1 \) into the quadratic equation to find \( k \).

#### For \( m = 1 \):
Substituting \( m = 1 \):
\[ 1^2 - 1 - k = 0 \]
This simplifies to:
\[ 1 - 1 - k = 0 \implies k = 0 \]

#### For \( m = -1 \):
Substituting \( m = -1 \):
\[ (-1)^2 - (-1) - k = 0 \]
This simplifies to:
\[ 1 + 1 - k = 0 \implies k = 2 \]

### Step 5: Conclusion
Thus, the values of \( k \) that satisfy the condition are \( k = 0 \) and \( k = 2 \).

### Final Answer
The values of \( k \) are \( k = 0 \) and \( k = 2 \).

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