Problem 67 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ x^{2}-4 y^{2}=1 $$

Step-by-Step Solution

Answer

The surface is a hyperbola centered at the origin, with vertices \((1,0)\) and \((-1,0)\), and asymptotes \(y=2x\) and \(y=-2x\).

Step 1: Identify the Type of Surface

The equation given is in the form of \( x^2 - 4y^2 = 1 \), which resembles the standard form of a hyperbola. This is a hyperbola centered along the \( x \)-axis.

Step 2: Compare with Standard Form

Recall the standard form of a hyperbola along the \( x \)-axis: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). By comparing, we have \( a^2 = 1 \) so \( a = 1 \), and \( b^2 = 4 \) so \( b = 2 \).

Step 3: Determine the Asymptotes

For hyperbolas of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equations of the asymptotes are \( y = \pm \frac{b}{a}x \). Substituting \( a = 1 \) and \( b = 2 \), the asymptotes are \( y = 2x \) and \( y = -2x \).

Step 4: Plot the Hyperbola

The center of the hyperbola is at the origin \((0,0)\). Plot the vertices at \((1, 0)\) and \((-1, 0)\) on the \( x \)-axis, since these are where the hyperbola meets the \( x \)-axis. Using the asymptotes and vertices, you can sketch the two branches of the hyperbola opening left and right, approaching the asymptotes.

Key Concepts

Asymptotes

Asymptotes are vital when you study hyperbolas. These invisible lines guide the curves of the hyperbola. Hyperbolas approach these lines but never touch them. Asymptotes provide a framework that helps predict the hyperbola's shape.

For a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes can be found using the formula \( y = \pm \frac{b}{a}x \). This is because the hyperbola shifts shape at its steepest point, determined by the ratio \( \frac{b}{a} \). In our example, the asymptotes are \( y = 2x \) and \( y = -2x \).

This means the curves of the hyperbola will get closer and closer to these lines as the values of \( x \) increase, but they will never intersect or cross these stigmas of direction. They help draw the hyperbola accurately by providing boundaries for its wings.

Standard Form of Hyperbola

The standard form of a hyperbola is crucial for identifying and graphing its shape. It sets the groundwork from which all measurements and analyses arise.

When a hyperbola is oriented along the \( x \)-axis, the standard form looks like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \( a^2 \) and \( b^2 \) are parameters that define the size and direction of the hyperbola. For the equation \( x^2 - 4y^2 = 1 \), comparing it to the standard form shows that \( a^2 = 1 \rightarrow a = 1 \) and \( b^2 = 4 \rightarrow b = 2 \).

This standard form helps identify the vertices of the hyperbola at \((a, 0)\) and \((-a, 0)\) for the \( x \)-axis oriented hyperbola. Knowing this form allows you to understand both vertical and horizontal hyperbolas and adjust the parameters \(a\) and \(b\) to change its shape.

Conic Sections

Conic sections classify curves like circles, ellipses, parabolas, and hyperbolas. These are formed by intersecting a plane with a cone at different angles.

Hyperbolas are among these fascinating shapes. When the intersecting plane is at a steep angle to the cone's base, two separate curves are formed, known as a hyperbola. This means the plane cuts through both nappes of the cone.

Exploring hyperbolas within conic sections unveils how manipulation of the cones can influence the resulting curves. Each conic section exhibits unique properties and applications, making them essential in geometry. For example, hyperbolas often model scenarios in physics and astronomy, like the paths of objects in gravitational fields or the shape of certain satellite dishes.