Problem 42 - Algebra 2

Question

ACT/SAT The foci of the graph are at \((\sqrt{13}, 0)\) and \((-\sqrt{13}, 0) .\) Which equation does the graph represent? $$ \begin{array}{l}{\text { A } \frac{x^{2}}{9}-\frac{y^{2}}{4}=1} \\ {\text { B } \frac{x^{2}}{3}-\frac{y^{2}}{2}=1} \\ {\text { C } \frac{x^{2}}{3}-\frac{y^{2}}{\sqrt{13}}=1} \\ {\text { D } \frac{x^{2}}{9}-\frac{y^{2}}{13}=1}\end{array} $$

Step-by-Step Solution

Answer

Option A: \(\frac{x^2}{9} - \frac{y^2}{4} = 1\) is the correct equation.

Step 1: Identify the type of conic section

Since the foci are located at \((\sqrt{13}, 0)\) and \((-\sqrt{13}, 0)\), which are along the x-axis, the conic section is a hyperbola centered at the origin. Hyperbolas have horizontal foci when the equation is in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Determine the distance c from center to each focus

In a hyperbola, the distance from the center to each focus is given by \(c\). From the given foci \((\sqrt{13}, 0)\) and \((-\sqrt{13}, 0)\), we have \(c = \sqrt{13}\).

Step 3: Recall the relationship between a, b, and c

For a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the relationship between \(a\), \(b\), and \(c\) is \(c^2 = a^2 + b^2\).

Step 4: Compare given options

Each option represents a hyperbola equation: - (A) \(\frac{x^2}{9} - \frac{y^2}{4} = 1\) with \(a^2 = 9\), \(b^2 = 4\), resulting in \(c^2 = 13\). - (B) \(\frac{x^2}{3} - \frac{y^2}{2} = 1\) leads to \(c^2 = 5\), incorrect.- (C) \(\frac{x^2}{3} - \frac{y^2}{\sqrt{13}} = 1\) yields an invalid \(c^2 = 13\).- (D) \(\frac{x^2}{9} - \frac{y^2}{13} = 1\) results in \(c^2 = 22\), incorrect.

Step 5: Formulate the solution

From the comparison, option (A) with \(a^2 = 9\), \(b^2 = 4\) satisfies \(c^2 = 13\) correctly based on the given focal equation of the hyperbola. Thus, the correct equation is \(\frac{x^2}{9} - \frac{y^2}{4} = 1\).

Key Concepts

Hyperbola

A hyperbola is one of the four conic sections, along with the circle, ellipse, and parabola. It is defined as a set of points where the difference of the distances to two fixed points, called the foci, is constant. In simpler terms, if you think of stretching a piece of string around two pins on a board and tracing out a curve, this curve could be one branch of a hyperbola.
Hyperbolas have two disconnected curves, known as branches, which open outward either horizontally or vertically. The orientation of the branches depends on the axis along which the foci lie:

• Horizontal hyperbolas have their foci along the x-axis and the equation typically starts with \(rac{x^2}{a^2} - rac{y^2}{b^2} = 1\).
• Vertical hyperbolas have their foci along the y-axis and use \(rac{y^2}{b^2} - rac{x^2}{a^2} = 1\).

Both types are symmetrical, with their own central intersection point known as the center of the hyperbola. Understanding hyperbolas is key to solving problems related to their properties and equations.

Foci

The foci (singular: focus) are essential in identifying and understanding hyperbolas. These points influence the overall shape and orientation of the hyperbola. When studying a hyperbola, it is important to know:

• The foci are always located symmetrically on the axis defined by the orientation of the hyperbola.
• In a standard equation \(rac{x^2}{a^2} - rac{y^2}{b^2} = 1\) or \( rac{y^2}{b^2} - rac{x^2}{a^2} = 1\), the distance from the center to each focus is denoted by \(c\).
• The distance \(c\) is critical because it describes how stretched out the hyperbola is relative to its center.

For example, with the foci given as \(\sqrt{13}, 0\) and \(-\sqrt{13}, 0\), it means they are placed on the x-axis and the hyperbola opens horizontally. Understanding the position of the foci helps in determining the correct equation of the hyperbola, as the distance between them, expressed as \(c = \sqrt{13}\), relates directly to the equation terms.

Equation of a Hyperbola

The equation of a hyperbola can look daunting, but breaking it down helps to decode its meaning and significance. A standard hyperbola equation comes in one of two major forms, based on the axis orientation:

• Horizontal: \(rac{x^2}{a^2} - rac{y^2}{b^2} = 1\)
• Vertical: \(rac{y^2}{b^2} - rac{x^2}{a^2} = 1\)

Let's dissect the components:

• The terms \(a^2\) and \(b^2\) are part of the denominators for the x and y terms respectively, dictating the directions and dimensions of the hyperbola's axes.
• The relationship \(c^2 = a^2 + b^2\) connects these values, where \(c\) represents the distance to each focus and helps in solving these equations specifically.
• The values of \(a\) and \(b\) determine how "stretched" or "compressed" the hyperbola's branches are.

In context, choosing the correct form based on given foci is vital. By assessing options against the rule \(c^2 = a^2 + b^2\), it is possible to find the right equation that satisfies the focal distance relationship. This process involves solving for which configuration of \(a^2\) and \(b^2\) accuratly describes the specific hyperbola.