Problem 8 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}-y^{2}=1$$

Step-by-Step Solution

Answer

The equation \(x^2 - y^2 = 1\) represents a hyperbola.

Step 1: Identify the General Form

The given equation is \(x^{2} - y^{2} = 1\). This equation can be compared to the general equation of a conic section in the form \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\). We can see that in this case, \(A = 1\), \(B = 0\), \(C = -1\), and the other coefficients are zero.

Step 2: Determine Type of Conic Section

We use the discriminant \(B^2 - 4AC\) to determine the type of conic. For \(x^2 - y^2 = 1\), with \(A = 1\), \(B = 0\), and \(C = -1\), the discriminant is \(0^2 - 4(1)(-1) = 4\). Because the discriminant is greater than zero \((4 > 0)\), the conic section is a hyperbola.

Key Concepts

Hyperbola

A hyperbola is one of the four types of conic sections, the others being ellipse, parabola, and circle. Hyperbolas can be visualized as two identical curves facing away from each other. These curves are called branches of the hyperbola.
Unlike circles and ellipses, hyperbolas are defined by subtraction in their equations. In our given equation, \(x^2 - y^2 = 1\), we see that one of the terms is negative: \(-y^2\). This negative sign is a key indicator of a hyperbola.

• Hyperbolas have a center, asymptotes, and vertices.
• They open either horizontally or vertically.
• The equation can be written in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\).

An easy way to distinguish hyperbolas is by using the discriminant. As explained in the original solution, a positive discriminant (greater than zero) indicates a hyperbola.

Discriminant

The discriminant is a handy tool for determining the type of conic without graphing it. It appears in the formula \(B^2 - 4AC\), derived from the general form of a conic equation \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\).
For the equation \(x^2 - y^2 = 1\), we have \(A = 1\), \(B = 0\), and \(C = -1\). Plugging these into the discriminant gives: \(0^2 - 4(1)(-1) = 4\). Since the result is positive (4 > 0), this confirms the graph is a hyperbola.

• Discriminant greater than zero: Hyperbola.
• Discriminant equals zero: Parabola.
• Discriminant less than zero: Ellipse or Circle.

This simple calculation saves time and helps avoid plotting points unnecessarily.

General Equation of a Conic

Conic sections are defined by equations that fit into the general form: \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\). Based on the coefficients, you can decide the type of the conic.
In the specific equation \(x^{2} - y^{2} = 1\), the general form takes shape as \(A = 1\), \(B = 0\), \(C = -1\), with \(D\), \(E\), and \(F\) being zero. This simplifies the identification process.

• If \(B = 0\), and \(A\) and \(C\) have opposite signs, it's a hyperbola.
• If \(A = C\) and \(B = 0\), it's a circle.
• If \(A \, \text{or} \, C = 0\) but not both, it's a parabola.

Recognizing the contribution of each coefficient helps in understanding the nature of the curve. Through this form, we learn not only what the conic looks like, but also its location and orientation.