The eccentricity of the hyperbola `-(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is given by

Options

A. `e=sqrt((a^(2)+b^(2))/(a^(2)))`
B. `e=sqrt((a^(2)-b^(2))/(a^(2)))`
C. `e=sqrt((b^(2)-a^(2))/(a^(2)))`
D. `e=sqrt((a^(2)-b^(2))/(b^(2)))`

Answer

To find the eccentricity of the hyperbola given by the equation

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

we will follow these steps:

### Step 1: Rewrite the equation in standard form

The given equation can be rearranged to match the standard form of a hyperbola. The standard form of a hyperbola is

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

To convert the given equation into this form, we multiply through by -1:

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

### Step 2: Identify the values of \(a\) and \(b\)

From the standard form, we can see that:

- \(a^2\) corresponds to the denominator of the \(x^2\) term, which is \(a^2\).
- \(b^2\) corresponds to the denominator of the \(y^2\) term, which is \(b^2\).

### Step 3: Use the formula for eccentricity

The formula for the eccentricity \(e\) of a hyperbola in standard form \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) is given by:

\[ e = \sqrt{1 + \frac{b^2}{a^2}}. \]

### Step 4: Substitute the values into the formula

Substituting the values of \(a\) and \(b\) into the formula, we have:

\[ e = \sqrt{1 + \frac{b^2}{a^2}}. \]

### Step 5: Final expression for eccentricity

Thus, the eccentricity of the hyperbola given by the equation

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

is

\[ e = \sqrt{1 + \frac{b^2}{a^2}}. \]