The equation of conjugate hyperbola of `x^2/8 - 3y^2/8 =1`

Options

A. `x^2- 3y^2 - 2x +8=0`
B. `3x^2- y^-2 - 2y -8=0`
C. `x^2- 3y^2 - 2x +10=0`
D. `x^2- 3y^2 =-8`

Answer

To find the equation of the conjugate hyperbola of the given equation \( \frac{x^2}{8} - \frac{3y^2}{8} = 1 \), we can follow these steps:

### Step 1: Identify the standard form of the hyperbola
The given equation is in the form:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \( a^2 = 8 \) and \( b^2 = \frac{8}{3} \).

### Step 2: Write down the conjugate hyperbola equation
The conjugate hyperbola of the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \]
Thus, substituting \( a^2 = 8 \) and \( b^2 = \frac{8}{3} \), we get:
\[ \frac{x^2}{8} - \frac{3y^2}{8} = -1 \]

### Step 3: Rearranging the equation
To rearrange the equation, we multiply through by -1:
\[ -\left(\frac{x^2}{8} - \frac{3y^2}{8}\right) = 1 \]
This simplifies to:
\[ -\frac{x^2}{8} + \frac{3y^2}{8} = 1 \]
or
\[ \frac{3y^2}{8} - \frac{x^2}{8} = 1 \]

### Step 4: Clear the denominators
To eliminate the fractions, multiply the entire equation by 8:
\[ 3y^2 - x^2 = 8 \]

### Step 5: Rearranging to standard form
Rearranging gives us the final equation of the conjugate hyperbola:
\[ x^2 - 3y^2 = -8 \]

### Final Answer
Thus, the equation of the conjugate hyperbola is:
\[ x^2 - 3y^2 = -8 \]

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