If one of the lines given by the equation `ax^(2)+6xy+by^(2)=0` bisects the angle between the co-ordinate axes, then value of `(a+b)` can be :

Options

A. -6
B. 3
C. 6
D. 12

Answer

To solve the problem, we need to find the value of \( a + b \) given that one of the lines represented by the equation \( ax^2 + 6xy + by^2 = 0 \) bisects the angle between the coordinate axes.

### Step 1: Understand the equation of the lines
The equation \( ax^2 + 6xy + by^2 = 0 \) represents a pair of straight lines through the origin. The lines can be expressed in the form \( y = mx \), where \( m \) is the slope.

### Step 2: Identify the lines that bisect the angle between the coordinate axes
The lines that bisect the angle between the coordinate axes are given by \( y = x \) and \( y = -x \).

### Step 3: Substitute \( y = x \) into the equation
Let’s first consider the case where one of the lines is \( y = x \). We substitute \( y = x \) into the equation:

\[ ax^2 + 6x(x) + bx^2 = 0 \]

This simplifies to:

\[ ax^2 + 6x^2 + bx^2 = 0 \]

Combining like terms gives:

\[ (a + b + 6)x^2 = 0 \]

### Step 4: Set the equation to zero
Since \( x^2 \) cannot be zero (as it represents a line through the origin), we set the coefficient to zero:

\[ a + b + 6 = 0 \]

This leads to:

\[ a + b = -6 \quad \text{(1)} \]

### Step 5: Substitute \( y = -x \) into the equation
Now, let’s consider the case where the line is \( y = -x \). We substitute \( y = -x \) into the equation:

\[ ax^2 + 6x(-x) + b(-x)^2 = 0 \]

This simplifies to:

\[ ax^2 - 6x^2 + bx^2 = 0 \]

Combining like terms gives:

\[ (a + b - 6)x^2 = 0 \]

### Step 6: Set the equation to zero
Again, since \( x^2 \) cannot be zero, we set the coefficient to zero:

\[ a + b - 6 = 0 \]

This leads to:

\[ a + b = 6 \quad \text{(2)} \]

### Step 7: Conclusion
From equations (1) and (2), we have:

1. \( a + b = -6 \)
2. \( a + b = 6 \)

These two equations cannot be true simultaneously unless we have a contradiction. Thus, we conclude that the possible values for \( a + b \) can be either \( -6 \) or \( 6 \) depending on which line bisects the angle.

### Final Answer
The possible values of \( a + b \) can be \( -6 \) or \( 6 \).

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