if the eccentricity of the hyperbola is `sqrt2`. then find the general equation of hyperbola.

Answer

To find the general equation of a hyperbola given that its eccentricity \( e \) is \( \sqrt{2} \), we can follow these steps:

### Step 1: Recall the formula for eccentricity of a hyperbola
The eccentricity \( e \) of a hyperbola is given by the formula:
\[ e = \frac{\sqrt{a^2 + b^2}}{a} \]
where \( a \) and \( b \) are the semi-major and semi-minor axes respectively.

### Step 2: Substitute the value of eccentricity
Given that \( e = \sqrt{2} \), we can substitute this into the eccentricity formula:
\[ \sqrt{2} = \frac{\sqrt{a^2 + b^2}}{a} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ \sqrt{2}a = \sqrt{a^2 + b^2} \]

### Step 4: Square both sides to remove the square root
Squaring both sides results in:
\[ 2a^2 = a^2 + b^2 \]

### Step 5: Rearrange the equation to find a relationship between \( a^2 \) and \( b^2 \)
Rearranging the equation gives:
\[ 2a^2 - a^2 = b^2 \implies a^2 = b^2 \]

### Step 6: Write the general equation of the hyperbola
The standard equation of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Since we found that \( a^2 = b^2 \), we can denote \( a^2 \) as \( k \) (where \( k \) is a positive constant). Thus, the equation becomes:
\[ \frac{x^2}{k} - \frac{y^2}{k} = 1 \]

### Step 7: Simplify the equation
This simplifies to:
\[ x^2 - y^2 = k \]

### Step 8: General form of the hyperbola
Thus, the general equation of the hyperbola can be expressed as:
\[ x^2 - y^2 = a^2 \quad \text{(where } a^2 = k \text{)} \]