53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1

Verified step by step guidance

1. Identify the type of conic section by comparing the given equation \(x^{2} - \frac{y^{2}}{2} = 1\) to the standard forms. Since it has a positive \(x^{2}\) term and a negative \(y^{2}\) term, it represents a hyperbola.
2. Rewrite the equation in the standard form of a hyperbola centered at the origin: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\). From the given equation, \(a^{2} = 1\) and \(b^{2} = 2\).
3. Determine the vertices, which lie along the transverse axis (the \(x\)-axis here). The vertices are located at \((\pm a, 0)\), so they are at \((\pm 1, 0)\).
4. Calculate the foci using the relationship \(c^{2} = a^{2} + b^{2}\). Find \(c\) and then the foci are at \((\pm c, 0)\).
5. Find the equations of the directrices, which are vertical lines given by \(x = \pm \frac{a^{2}}{c}\). Substitute the values of \(a^{2}\) and \(c\) to write the directrix equations.

Key Concepts

Standard Form of Hyperbola

A hyperbola is defined by an equation in the form (x²/a²) - (y²/b²) = 1 or (y²/a²) - (x²/b²) = 1. Recognizing the standard form helps identify the orientation (horizontal or vertical) and the values of a² and b², which are essential for locating vertices, foci, and directrices.

Foci and Vertices of a Hyperbola

Vertices are points where the hyperbola intersects its principal axis, located at ±a from the center. Foci lie further along the same axis at a distance c, where c² = a² + b². These points define the shape and key properties of the hyperbola.

Directrices of a Hyperbola

Directrices are fixed lines used to define a hyperbola via the ratio of distances to a focus and to a directrix (eccentricity). For a hyperbola, directrices are located at a distance of a/e from the center, where e = c/a is the eccentricity, and help in understanding its geometric properties.