Find the equation of hyperbola which passes through the point (-3,2) and its conjugate axis is 8.

Answer

To find the equation of the hyperbola that passes through the point (-3, 2) and has a conjugate axis of length 8, we can follow these steps:

### Step 1: Determine the value of \( b \)
The length of the conjugate axis of a hyperbola is given by \( 2b \). Since the conjugate axis is 8, we can set up the equation:

\[ 2b = 8 \]

Dividing both sides by 2 gives:

\[ b = 4 \]

### Step 2: Write the general equation of the hyperbola
The standard form of the equation of a hyperbola with a horizontal transverse axis is:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Substituting \( b = 4 \) into the equation gives:

\[ \frac{x^2}{a^2} - \frac{y^2}{16} = 1 \]

### Step 3: Substitute the point (-3, 2) into the equation
Since the hyperbola passes through the point (-3, 2), we can substitute \( x = -3 \) and \( y = 2 \) into the equation:

\[ \frac{(-3)^2}{a^2} - \frac{(2)^2}{16} = 1 \]

This simplifies to:

\[ \frac{9}{a^2} - \frac{4}{16} = 1 \]

### Step 4: Simplify the equation
We know that \( \frac{4}{16} = \frac{1}{4} \), so we can rewrite the equation as:

\[ \frac{9}{a^2} - \frac{1}{4} = 1 \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply through by \( 4a^2 \):

\[ 4a^2 \left( \frac{9}{a^2} \right) - 4a^2 \left( \frac{1}{4} \right) = 4a^2(1) \]

This simplifies to:

\[ 36 - a^2 = 4a^2 \]

### Step 6: Solve for \( a^2 \)
Rearranging the equation gives:

\[ 36 = 5a^2 \]

Dividing both sides by 5, we find:

\[ a^2 = \frac{36}{5} \]

### Step 7: Write the final equation of the hyperbola
Now that we have \( a^2 \) and \( b^2 \), we can substitute these values back into the hyperbola equation:

\[ \frac{x^2}{\frac{36}{5}} - \frac{y^2}{16} = 1 \]

To express this in a more standard form, we can multiply through by 80 (the least common multiple of 5 and 16):

\[ 80 \left( \frac{x^2}{\frac{36}{5}} \right) - 80 \left( \frac{y^2}{16} \right) = 80 \]

This simplifies to:

\[ \frac{400x^2}{36} - 5y^2 = 80 \]

Thus, the final equation of the hyperbola is:

\[ \frac{100x^2}{9} - 5y^2 = 80 \]