What is the eccentricity of rectangular hyperbola?

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. `sqrt(5)`
D. `sqrt(6)`

Answer

To find the eccentricity of a rectangular hyperbola, we can follow these steps:

### Step 1: Understand the equation of a rectangular hyperbola
The standard equation of a rectangular hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
For a rectangular hyperbola, the condition is \(a = b\).

### Step 2: Substitute \(a = b\) into the equation
Since we know that for a rectangular hyperbola \(a = b\), we can substitute \(b\) with \(a\) in the equation:
\[ \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \]

### Step 3: Simplify the equation
This simplifies to:
\[ \frac{x^2 - y^2}{a^2} = 1 \]
Multiplying both sides by \(a^2\) gives:
\[ x^2 - y^2 = a^2 \]

### Step 4: Use the formula for eccentricity
The formula for the eccentricity \(e\) of a hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Since \(a = b\), we can substitute \(b\) with \(a\):
\[ e = \sqrt{1 + \frac{a^2}{a^2}} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 5: Determine the value of eccentricity
Since eccentricity is always positive, we take the positive root:
\[ e = \sqrt{2} \]

### Conclusion
Thus, the eccentricity of a rectangular hyperbola is:
\[ e = \sqrt{2} \]

---