Problem 34 - Thomas Calculus

Question

Exercises \(27-34\) give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch. $$ 64 x^{2}-36 y^{2}=2304 $$

Step-by-Step Solution

Answer

The hyperbola's standard form is \(\frac{x^2}{36} - \frac{y^2}{64} = 1\). Asymptotes are \(y = \pm\frac{4}{3}x\), and foci are at (±10, 0).

Step 1: Identify the Hyperbola Equation

We start with the equation \(64x^2 - 36y^2 = 2304\), which is the equation of a hyperbola in a general form. Our task is to convert it into the standard form.

Step 2: Divide to Simplify

To simplify the equation into the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), divide the entire equation by 2304:\[\frac{64x^2}{2304} - \frac{36y^2}{2304} = \frac{2304}{2304}\]This simplifies to:\[\frac{x^2}{36} - \frac{y^2}{64} = 1\].

Step 3: Identify Parameters

Here we identify the squared terms:- \(a^2 = 36\) so \(a = 6\)- \(b^2 = 64\) so \(b = 8\)The standard form of the hyperbola is \(\frac{x^2}{6^2} - \frac{y^2}{8^2} = 1\), which indicates a horizontal hyperbola because \(x^2\) is over the larger denominator.

Step 4: Find Asymptotes

For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are given by:\[y = \pm\frac{b}{a}x\]Substitute the values of \(a = 6\) and \(b = 8\):\[y = \pm\frac{8}{6}x\]Simplifying gives \(y = \pm\frac{4}{3}x\).

Step 5: Consider Hyperbola's Foci

The distance to the foci \(c\) is given by \(c = \sqrt{a^2 + b^2}\).\[c = \sqrt{36 + 64} = \sqrt{100} = 10\]The foci are located at \((\pm c, 0) = (\pm 10, 0)\).

Step 6: Sketch the Hyperbola

To sketch the hyperbola, plot the center at the origin (0,0), plot vertices at (±6, 0), and draw the rectangle centered at the origin extending horizontally to ±6 and vertically to ±8. Draw the asymptotes \(y = \pm\frac{4}{3}x\) passing through the corners of this rectangle. Finally, draw the hyperbola opening to the sides along the x-axis, approaching the asymptotes, and plot foci at (±10, 0).

Key Concepts

Standard Form of Hyperbola

The standard form of a hyperbola provides a clear and structured way to represent the hyperbola equation. In mathematics, the standard form for a horizontal hyperbola is given by:

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

This form is useful because it directly tells us about the size and orientation of the hyperbola.
For instance, in the example provided, converting the equation \(64x^2 - 36y^2 = 2304\) into this standard form helps us to identify important characteristics such as whether the hyperbola is horizontal or vertical.
By dividing each term by 2304, we simplify it to \(\frac{x^2}{36} - \frac{y^2}{64} = 1\), indicating a horizontal hyperbola due to the larger denominator being associated with \(y^2\).
Knowing how to rewrite the equation into this form is crucial for further identifying other properties like vertices, asymptotes, and foci.

Hyperbola Asymptotes

Asymptotes are lines that a hyperbola approaches but never actually touches or crosses. They play a crucial role in helping us sketch the shape of a hyperbola and predict how it behaves as the values of x and y increase.
For hyperbolas in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptote equations are given by:

• \(y = \pm\frac{b}{a}x\)

In our example, substituting \(a = 6\) and \(b = 8\) gives us the equations \(y = \pm\frac{4}{3}x\). This tells us that the asymptotes slope at an angle of \(\pm\frac{4}{3}\), guiding the direction of the branches of the hyperbola as they extend outwards.
The asymptotes meet at the center of the hyperbola, and they help in forming a reference frame for drawing the hyperbola itself. By sketching the asymptotes, one gains a visual cue on how the hyperbola will "hug" these lines at infinity, creating the classical open-ended curve.

Hyperbola Foci

A hyperbola has two foci, which are significant points that are used in the definition and construction of the curve. The peculiar nature of a hyperbola means that the difference in the distances from these foci to any point on the hyperbola is constant.
For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the distance to each focus \(c\) is determined by the formula:

• \(c = \sqrt{a^2 + b^2}\)

Calculating this in our example with \(a = 6\) and \(b = 8\), we find \(c = \sqrt{36 + 64} = 10\). This tells us that the foci are situated at \((\pm 10, 0)\).
These points lie along the transverse axis of the hyperbola, which is in the direction that the hyperbola opens. Locating the foci is essential when sketching the hyperbola, as they help ensure accuracy in the curve's shape and orientation.

Conic Sections

Hyperbolas are part of a group of curves known as conic sections. These shapes arise by intersecting a plane with a double-napped cone at different angles and positions. The main types of conic sections are:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

What makes hyperbolas unique among conic sections is their open curve structure, consisting of two distinct branches. This happens when the intersecting plane cuts both naps of the cone.
Conic sections like hyperbolas can describe a wide range of phenomena in nature and are often seen in the paths of celestial objects or in certain types of waves and lenses.
Understanding hyperbolas as part of these conic sections allows for a broader comprehension of geometry and its application across various scientific fields.