Problem 66 - Algebra 2

Question

Find the foci of each hyperbola. Draw the graph. \(\frac{x^{2}}{49}-\frac{y^{2}}{36}=1\)

Step-by-Step Solution

Answer

The foci of the hyperbola are at \((-\sqrt{85}, 0)\) and \((\sqrt{85}, 0)\).

Step 1: Identify the values of a and b

From the equation \(\frac{x^{2}}{49} - \(\frac{y^{2}}{36} = 1, it's clear that \(a^{2} = 49\) and \(b^{2} = 36\). Therefore, \(a = 7\) and \(b = 6\).

Step 2: Compute the value of c

Use the formula \(c = \sqrt{a^{2} + b^{2}}\) to compute the value of c. Substituting \(a = 7\) and \(b = 6\) into the formula gives \(c = \sqrt{49 + 36} = \sqrt{85}\).

Step 3: Find the foci

The foci of the hyperbola are at \((\pm c, 0)\), so substituting \(c = \sqrt{85}\) into the formula gives the foci as \((-\sqrt{85}, 0)\) and \((\sqrt{85}, 0)\).

Step 4: Draw the graph

Draw the hyperbola on an x-y graph. The center of the hyperbola is at the origin. Draw the two branches of the hyperbola passing through the foci, which were computed as \((-\sqrt{85}, 0)\) and \((\sqrt{85}, 0)\). The branches should open towards the positive and negative x-directions respectively, with the foci as the furthest points in each direction.

Key Concepts

Foci of Hyperbola

The foci of a hyperbola are vital reference points inside its shape, similar to the focus of a parabola. But unlike a parabola, a hyperbola has two foci. In the case of a hyperbola centered at the origin, the foci lie along the axis that corresponds to the positive term in the equation. For the equation \(\frac{x^{2}}{49} - \frac{y^{2}}{36} = 1\), the positive term involves \(x\), so the foci will be horizontally aligned.

To find the foci, we use the formula \(c = \sqrt{a^{2} + b^{2}}\), where \(a\) and \(b\) are derived from the denominators of the equation's respective terms. Here, \(a^{2} = 49\), and \(b^{2} = 36\), giving \(a = 7\) and \(b = 6\). Substituting these into the formula results in \(c = \sqrt{85}\). Hence, the foci are located at \((\pm \sqrt{85}, 0)\).

This process of calculating \(c\) ensures that each hyperbola's foci are precisely located for reliable graphing and further study, reinforcing their importance in defining the overall shape of the hyperbola.

Equation of Hyperbola

The equation of a hyperbola reflects its opening direction and dimensions. General hyperbola equations take two forms depending on the transverse orientation:

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

In our exercise, the hyperbola \(\frac{x^{2}}{49} - \frac{y^{2}}{36} = 1\) is horizontal as the positive term is \(x^{2}\).

This equation tells us crucial characteristics about the hyperbola:

• The center is at the origin \((0,0)\).
• The distances \(a\) and \(b\) describe the spread and slant of the hyperbola from the center along the axes.

Once you know \(a\) and \(b\), deriving \(c\) through \(c = \sqrt{a^{2} + b^{2}}\) sets the foci position. Understanding the equation gives you the information needed to accurately draw and analyze the hyperbola's properties.

Graphing Hyperbolas

Graphing a hyperbola starts with understanding its equation and finding its core components: the center, foci, and asymptotes. Begin with marking the center at the origin \((0,0)\), and place the foci at \((\pm\sqrt{85}, 0)\) based on your calculations.

Next, sketch the asymptotes. For a horizontal hyperbola like \(\frac{x^{2}}{49} - \frac{y^{2}}{36} = 1\), the asymptotes are straight lines passing through the center with slopes \(\pm \frac{b}{a}\). Here, these slopes are \(\pm \frac{6}{7}\). Draw these diagonal lines extending in all directions.

The asymptotes guide the hyperbola's open sides. Sketch two branches of the hyperbola moving outward from the center, approaching but never crossing these asymptotes. Remember, these branches indicate the range of points twice the distance from a focus than to the corresponding directrix.

The sketch should depict two distinct, mirror-image curves that open along the axis of the primary variable \(x\), emphasizing the hyperbola's horizontal nature in this context.