Problem 12 - Algebra and Trigonometry

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ \frac{x^{2}}{2}-y^{2}=1 $$

Step-by-Step Solution

Answer

Vertices: \((\sqrt{2}, 0), (-\sqrt{2}, 0)\); Foci: \((\sqrt{3}, 0), (-\sqrt{3}, 0)\); Asymptotes: \(y = \pm \frac{\sqrt{2}}{2}x\).

Step 1: Identify the Equation Form

The given hyperbola equation is \( \frac{x^2}{2} - y^2 = 1 \). This hyperbola is in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \( a^2 = 2 \) and \( b^2 = 1 \). Thus, \( a = \sqrt{2} \) and \( b = 1 \). This form indicates that the hyperbola opens horizontally.

Step 2: Determine the Vertices

For a hyperbola that opens horizontally, the vertices are located at \( (\pm a, 0) \). Since \( a = \sqrt{2} \), the vertices are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).

Step 3: Find the Foci

The distance between the center and each focus of a hyperbola is \( c \), where \( c^2 = a^2 + b^2 \). Therefore, \( c^2 = 2 + 1 = 3 \) and \( c = \sqrt{3} \). Hence, the foci are at \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \).

Step 4: Determine the Asymptotes

The equations of the asymptotes for a horizontally opening hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are \( y = \pm \frac{b}{a} x \). Thus, the asymptotes for this hyperbola are \( y = \pm \frac{1}{\sqrt{2}} x \), which simplifies to \( y = \pm \frac{\sqrt{2}}{2} x \).

Step 5: Sketch the Graph

Draw the vertices at \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \) on a coordinate plane. Then, plot the foci at \( (\sqrt{3}, 0) \) and \( (-\sqrt{3}, 0) \). Lightly draw the asymptotes \( y = \pm \frac{\sqrt{2}}{2} x \) as dashed lines. Sketch the hyperbola opening horizontally, approaching the asymptotes but never touching them.

Key Concepts

Vertices of a Hyperbola

Understanding the vertices of a hyperbola is fundamental to graphing and analyzing these fascinating curves. In the given hyperbola equation \( \frac{x^2}{2} - y^2 = 1 \), the vertices play a crucial role in determining the shape and orientation of the hyperbola.
The formula for a hyperbola that opens horizontally is given by \( x^2/a^2 - y^2/b^2 = 1 \). Here, the vertices are positioned along the x-axis at \((\pm a, 0)\). Calculating \(a\) from the equation, we found \( a = \sqrt{2} \).

• The first vertex is located at \((\sqrt{2}, 0)\), which is slightly more than one unit from the origin along the positive x-axis.
• The second vertex is at \((-\sqrt{2}, 0)\), a symmetrical counterpart on the negative x-axis.

These points are the extrema of the hyperbola in the horizontal direction, forming a natural outline as the hyperbola stretches outward.

Foci of a Hyperbola

The foci of a hyperbola are not just ordinary points; they are central to its definition and properties. For the hyperbola \( \frac{x^2}{2} - y^2 = 1 \), the foci help determine its stretched, vase-like appearance.
The calculation begins with determining \(c\), the distance from the center to each focus, using the relationship \( c^2 = a^2 + b^2 \). Here, \( a^2 = 2 \) and \( b^2 = 1 \) give us \( c^2 = 2 + 1 = 3 \), leading to \( c = \sqrt{3} \).

• One focus is located at \((\sqrt{3}, 0)\), a little more than beyond the vertex on the positive x-axis.
• The other focus is at \((-\sqrt{3}, 0)\), an equal distance from the center in the opposite direction.

The foci are deeper within the hyperbola than the vertices, influencing the hyperbola's curves as it extends toward infinity.

Asymptotes of a Hyperbola

Asymptotes are like invisible guides that the arms of a hyperbola follow as they extend further from the center. They don't intersect the hyperbola but shape its direction.
For the hyperbola given by \( \frac{x^2}{2} - y^2 = 1 \), we use the formula for asymptotes: \( y = \pm \frac{b}{a} x \). Here, \( b = 1 \) and \( a = \sqrt{2} \), which gives the slopes of the asymptotes \( \pm \frac{1}{\sqrt{2}} \).
This simplifies to the asymptote equations:

• \( y = \frac{\sqrt{2}}{2} x \)
• \( y = -\frac{\sqrt{2}}{2} x \)

These asymptotes appear as straight dashed lines through the origin, and their gentle slopes indicate the gradual opening of the hyperbola as it approaches them, providing a framework for its wider parts.