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The eccentricity of a hyperbola is a measure of how much it deviates from being circular. It is denoted by the letter \(e\). The formula used to calculate the eccentricity of a hyperbola is \(e = \sqrt{1 + \frac{b^2}{a^2}}\). The values of \(a\) and \(b\) are found from the equation of the hyperbola in its standard form, \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). For a hyperbola, \(e\) is always greater than 1.

In the exercise problem, \(a^2 = 1\) and \(b^2 = 1\). By substituting these into the formula we get \(e = \sqrt{1 + \frac{1}{1}} = \sqrt{2}\).

• The hyperbola's eccentricity is \(\sqrt{2}\).
• This value indicates that the hyperbola is more elongated compared to an ellipse, which has \(e < 1\).

The foci of a hyperbola are special points located along the transverse axis. They play a crucial role in its geometric properties. The distance from the center to each focus (denoted by \(c\)) is given by the formula \(c = \sqrt{a^2 + b^2}\).

For the given hyperbola \(x^2 - y^2 = 1\), substituting \(a^2 = 1\) and \(b^2 = 1\) into the formula gives \(c = \sqrt{2}\). Hence, the foci of this hyperbola are located at \((\pm \sqrt{2}, 0)\).

• The foci are on the x-axis because the equation has \(x^2\) first.
• The hyperbola has two foci: one at \((\sqrt{2}, 0)\) and another at \((-\sqrt{2}, 0)\).

Directrices of a hyperbola are lines that are perpendicular to the transverse axis. They work together with the foci to define the hyperbola. The equations for the directrices are derived using \(x = \pm \frac{a^2}{c}\), where \(c\) is the distance to the foci.

Using the values \(a^2 = 1\) and \(c = \sqrt{2}\), we substitute them into the formula to find that the directrices are \(x = \pm \frac{1}{\sqrt{2}}\).

• Directrices are vertical lines for this hyperbola.
• The lines \(x = \frac{1}{\sqrt{2}}\) and \(x = -\frac{1}{\sqrt{2}}\) mark the positions of the directrices.

The standard form of a hyperbola is a handy way to express its equation, allowing us to easily identify its attributes like its foci, vertices, and asymptotes. The form is either \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontally oriented hyperbola or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for a vertically oriented one.

In the provided exercise, the equation is \(x^2 - y^2 = 1\), which fits the standard form \(\frac{x^2}{1} - \frac{y^2}{1} = 1\). This tells us:

• '\(a^2 = 1\)' explains the horizontal orientation.
• '\(b^2 = 1\)' confirms the shape of the hyperbola is formed symmetrically along the axes.

With the standard form, identifying properties such as the eccentricity, foci, and directrices becomes straightforward. Keeping equations in this form ensures calculations and conceptual understanding are simplified.