If the centre of the hyperbola whose equation is `ax^(2) + 2hxy + by^(2) + 2gx + 2fy + c = 0 " be " (alpha, beta)` , then find the equation of the asymptotes.

Answer

To find the equation of the asymptotes of the hyperbola given by the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) with center at \( (\alpha, \beta) \), we can follow these steps:

### Step 1: Understand the equation of the hyperbola
The general form of the hyperbola is given by:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
This equation can represent a hyperbola that is rotated and translated from the origin.

### Step 2: Identify the center of the hyperbola
The center of the hyperbola is given as \( (\alpha, \beta) \). This means that we need to find the asymptotes of the hyperbola at this center.

### Step 3: Formulate the function \( f(x, y) \)
Define the function:
\[ f(x, y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c \]
The asymptotes can be derived from the condition that the function behaves like a combination of lines.

### Step 4: Set up the equation for asymptotes
The asymptotes can be found using the expression:
\[ f(x, y) - f(\alpha, \beta) = 0 \]
This means we need to evaluate \( f(\alpha, \beta) \):
\[ f(\alpha, \beta) = a\alpha^2 + 2h\alpha\beta + b\beta^2 + 2g\alpha + 2f\beta + c \]

### Step 5: Substitute into the asymptote equation
Now, substituting this back into our asymptote equation gives:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c - (a\alpha^2 + 2h\alpha\beta + b\beta^2 + 2g\alpha + 2f\beta + c) = 0 \]
The constant \( c \) cancels out:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy - (a\alpha^2 + 2h\alpha\beta + b\beta^2 + 2g\alpha + 2f\beta) = 0 \]

### Step 6: Simplify the equation
Let \( k = a\alpha^2 + 2h\alpha\beta + b\beta^2 + 2g\alpha + 2f\beta \). Thus, we can rewrite the equation of the asymptotes as:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy - k = 0 \]

### Step 7: Identify the asymptotes
The asymptotes of the hyperbola can be obtained by setting the quadratic terms to zero:
\[ ax^2 + 2hxy + by^2 = 0 \]
This represents the equations of the asymptotes.

### Final Equation of Asymptotes
To find the explicit equations of the asymptotes, we can factor the quadratic equation:
\[ ax^2 + 2hxy + by^2 = 0 \]
This can be solved using the quadratic formula or by finding the roots, leading to the equations of the asymptotes.