Problem 15 - Algebra and Trigonometry

Question

\(15-28=(a)\) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$ x y=8 $$

Step-by-Step Solution

Answer

(a) It's a hyperbola. (b) Using rotation, \(X^2-Y^2=8\) with \(\theta=\pi/4\). (c) Sketch intersects the axes at 45 degrees.

Step 1: Set Up the Equation

The given equation is \(xy = 8\). This is a hyperbola because it has a binary product term without squared terms.

Step 2: Convert to Standard Form

The equation is already in the standard form \(xy = 8\), indicating it's a rectangular hyperbola.

Step 3: Use a Rotation of Axes

To eliminate the \(xy\)-term, use the rotation transformation given by \(x = X\cos(\theta) - Y\sin(\theta)\) and \(y = X\sin(\theta) + Y\cos(\theta)\). For rectangular hyperbolas, \(\theta = \frac{\pi}{4}\) is used.This simplifies to:\[xy = (X\cos\left(\frac{\pi}{4}\right) - Y\sin\left(\frac{\pi}{4}\right))(X\sin\left(\frac{\pi}{4}\right) + Y\cos\left(\frac{\pi}{4}\right)) = X^2 - Y^2 = 8\]

Step 4: Sketch the Graph

The transformed equation \(X^2 - Y^2 = 8\) is a hyperbola centered at the origin with axes aligned along the coordinate axes. It opens towards the coordinate axes in the rotated system. Sketch two intersecting lines at 45-degree angles to the \(x\) and \(y\) axes. Rotate 45 degrees to align the hyperbola with the axes. Mark the asymptotes along the \(X\) and \(Y\) axes.

Key Concepts

Hyperbola

A hyperbola is a fascinating shape that comes from slicing a double cone (think of two ice cream cones touching tip to tip) diagonally. Specifically, when we talk about a hyperbola in mathematics, we often mean a set of points where the difference of distances from two fixed points, called foci, is constant. Hyperbolas have two distinct branches, often mirroring each other, and they open in slightly opposite directions.

• The standard form of a hyperbola with the origin as its center is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
• If the equation instead starts with \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), then the hyperbola opens up and down.
• For rectangular hyperbolas, like \(xy = 8\), the branches are symmetrical about the axes.

Hyperbolas occur not only in geometry, but also in various fields like physics and engineering, where they can describe the path of satellites or the cooling pattern of heated elements.

Rotation of Axes

Rotating the axes is a useful trick in geometry. It's like turning your paper around to look at a problem from a new angle. In mathematics, this is often used to simplify equations, especially to remove terms like \(xy\) that make solving harder.

The idea is to create new coordinate axes \(X\) and \(Y\) from old ones \(x\) and \(y\). We express these new axes in terms of trigonometric functions, which are like setting angles you might have with rotation:

• For \(x = X\cos(\theta) - Y\sin(\theta)\)
• And \(y = X\sin(\theta) + Y\cos(\theta)\)

For a rectangular hyperbola like \(xy = 8\), we choose \(\theta = \frac{\pi}{4}\), which is a 45-degree rotation. This choice simplifies our equation to something like \(X^2 - Y^2 = C\), aligning the hyperbola with the axes. This method allows you to easily see and understand the graph's shape by transforming it to a more typical and recognizable form.

Discriminant

The discriminant is a great tool to help you understand which conic section you're dealing with. It's like a magic number you calculate to identify the shape among ellipses, parabolas, or hyperbolas.
For any general quadratic equation in the form \(ax^2 + bxy + cy^2 + dx + ey + f = 0\), the discriminant formula is given by \(\Delta = b^2 - 4ac\). This tiny calculation holds the key to decoding the curve.

• If \(\Delta > 0\), it's a hyperbola.
• If \(\Delta = 0\), it becomes a parabola.
• If \(\Delta < 0\), you're looking at an ellipse.

In the problem of \(xy = 8\), you might start converting it to the form \(0x^2 + 1xy + 0y^2 -8 = 0\). Here, \(a = 0, b = 1, \) and \(c = 0\) reflect the structure. With this, \(\Delta = 1^2 - 4(0)(0) = 1 > 0\), confirming a hyperbola.