Find the equation of the hyperbola, whose asymptotes are the straight lines `(x + 2y + 3) = 0, (3x + 4y + 5) = 0` and which passes through the point (1-1).

Answer

To find the equation of the hyperbola whose asymptotes are given by the lines \(x + 2y + 3 = 0\) and \(3x + 4y + 5 = 0\) and which passes through the point \((1, -1)\), we can follow these steps:

### Step 1: Write the equations of the asymptotes
The asymptotes of the hyperbola are given as:
1. \(x + 2y + 3 = 0\)
2. \(3x + 4y + 5 = 0\)

### Step 2: Express the asymptotes in slope-intercept form
We can rewrite the equations of the asymptotes in the form \(y = mx + b\) to identify their slopes.

1. For the first asymptote:
\[ 2y = -x - 3 \implies y = -\frac{1}{2}x - \frac{3}{2} \]
(Slope \(m_1 = -\frac{1}{2}\))

2. For the second asymptote:
\[ 4y = -3x - 5 \implies y = -\frac{3}{4}x - \frac{5}{4} \]
(Slope \(m_2 = -\frac{3}{4}\))

### Step 3: Form the equation of the hyperbola
The general form of the equation of a hyperbola with given asymptotes can be expressed as:
\[ (x + 2y + 3)(3x + 4y + 5) + k = 0 \]
where \(k\) is a constant.

### Step 4: Expand the product
Now we will expand \((x + 2y + 3)(3x + 4y + 5)\):
\[ = x(3x + 4y + 5) + 2y(3x + 4y + 5) + 3(3x + 4y + 5) \]
\[ = 3x^2 + 4xy + 5x + 6xy + 8y^2 + 10y + 9x + 12y + 15 \]
Combining like terms:
\[ = 3x^2 + (4xy + 6xy) + 5x + 9x + (8y^2) + (10y + 12y) + 15 \]
\[ = 3x^2 + 10xy + 8y^2 + 14x + 22y + 15 \]

### Step 5: Substitute the point (1, -1)
Now we substitute the point \((1, -1)\) into the equation:
\[ 3(1)^2 + 10(1)(-1) + 8(-1)^2 + 14(1) + 22(-1) + 15 + k = 0 \]
Calculating each term:
\[ = 3 - 10 + 8 + 14 - 22 + 15 + k = 0 \]
\[ = 3 - 10 + 8 + 14 - 22 + 15 + k = 0 \]
\[ = 3 + 8 + 14 + 15 - 10 - 22 + k = 0 \]
\[ = 38 - 32 + k = 0 \]
\[ = 6 + k = 0 \implies k = -6 \]

### Step 6: Write the final equation of the hyperbola
Now we substitute \(k\) back into the equation:
\[ (x + 2y + 3)(3x + 4y + 5) - 6 = 0 \]
Thus, the equation of the hyperbola is:
\[ 3x^2 + 10xy + 8y^2 + 14x + 22y + 9 = 0 \]

### Final Answer
The equation of the hyperbola is:
\[ 3x^2 + 10xy + 8y^2 + 14x + 22y + 9 = 0 \]