Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). $$ 10 x^{2}-25 y^{2}=100 $$

Step-by-Step Solution

Answer

The hyperbola has vertices at \(( \pm \sqrt{10}, 0)\), foci at \(( \pm \sqrt{14}, 0)\), and asymptotes \( y = \pm \frac{\sqrt{10}}{5} x \).

Step 1: Rewrite the Equation in Standard Form

The given equation is \(10x^2 - 25y^2 = 100\). First, divide every term by 100 to simplify the equation: \[ \frac{10x^2}{100} - \frac{25y^2}{100} = \frac{100}{100} \] Simplifying gives: \[ \frac{x^2}{10} - \frac{y^2}{4} = 1 \] This represents the standard form of a hyperbola centered at the origin aligned with the x-axis.

Step 2: Identify the Center, Vertices, and Foci

For the hyperbola \( \frac{x^2}{10} - \frac{y^2}{4} = 1 \), the center is at the origin \((0, 0)\). - Vertices: Given \( a^2 = 10 \), the vertices (along the x-axis) are at \(( \pm \sqrt{10}, 0)\). - Foci: Use \( c^2 = a^2 + b^2 \) where \( b^2 = 4 \). Thus, \( c = \sqrt{14} \). The foci are at \(( \pm \sqrt{14}, 0)\).

Step 3: Find the Asymptotes

The equations of the asymptotes for a hyperbola centered at the origin \(\left(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\right)\) are given by \( y = \pm \frac{b}{a}x \). Substituting \( a^2 = 10 \) and \( b^2 = 4 \), we find \( \frac{b}{a} = \frac{2}{\sqrt{10}} \). Simplifying, the asymptotes are: \[ y = \pm \frac{\sqrt{10}}{5} x \]

Step 4: Sketch the Graph

To sketch the graph: - Draw the center at \((0, 0)\). - Mark the vertices at \(( \sqrt{10}, 0)\) and \(( -\sqrt{10}, 0)\). - Plot the foci at \(( \sqrt{14}, 0)\) and \(( -\sqrt{14}, 0)\). - Draw the asymptotes through the center with slopes \( \pm \frac{\sqrt{10}}{5} \). - Sketch the hyperbola opening along the x-axis and approaching the asymptotes.

Key Concepts

Vertices of a Hyperbola

In a hyperbola, the vertices are key points that lie on the major axis and help to define its shape and structure. For the hyperbola given in our exercise, the standard form of the equation is \( \frac{x^2}{10} - \frac{y^2}{4} = 1 \), which tells us it is centered at the origin \((0, 0)\) and opens horizontally.
The vertices are located at the endpoints of the hyperbola's major axis. The distance from the center to each vertex is represented by \(a\), where \(a^2\) is the denominator of the first term in the equation.

• Here, \(a^2 = 10\), so \(a = \sqrt{10}\).
• The vertices, therefore, are at the points \((\pm\sqrt{10}, 0)\).

These vertices indicate where each "branch" of the hyperbola intersects the x-axis. They are crucial for sketching an accurate graph, as the hyperbola will appear to expand outward from these points.

Foci of a Hyperbola

The foci (singular: focus) of a hyperbola are two special points located on the major axis outside of the vertices.
They are essential in defining the hyperbola itself because each point on the hyperbola has the property that the difference in distances to the two foci is constant.
To find the foci of the hyperbola, we use the formula \(c^2 = a^2 + b^2\).

• For our hyperbola \( \frac{x^2}{10} - \frac{y^2}{4} = 1\), we have \(a^2 = 10\) and \(b^2 = 4\).
• Thus, \(c^2 = 10 + 4 = 14\), giving \(c = \sqrt{14}\).
• This places the foci at the points \((\pm\sqrt{14}, 0)\).

Understanding the position of the foci helps when plotting the hyperbola, as they influence the hyperbola's shape and ensure the branches of the hyperbola extend towards these points.

Asymptotes of a Hyperbola

Asymptotes are invisible lines that a hyperbola approaches but never actually meets.
They provide direction and help set the angle at which the hyperbola's branches extend to infinity.
For a hyperbola given in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by \( y = \pm \frac{b}{a}x \).

• With \(a^2 = 10\) and \(b^2 = 4\), we have \(a = \sqrt{10}\) and \(b = 2\).
• Thus, the slope of the asymptotes is \(\frac{b}{a} = \frac{2}{\sqrt{10}} = \frac{\sqrt{10}}{5}\).

Therefore, the equations of the asymptotes are \( y = \pm \frac{\sqrt{10}}{5} x \).
These lines serve as guides for sketching the hyperbola, marking the "directions" in which each branch of the hyperbola opens and extends toward infinity.