Problem 15 - Precalculus Mathematics for Calculus

Question

(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 \(xy\)-term. (c) Sketch the graph. $$x y=8$$

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the coefficients

For the equation \(xy=8\), rearrange it as \(0x^2 + 0y^2 + 1xy - 8 = 0\). The coefficients are: \(A = 0\), \(B = 1\), \(C = 0\).

Step 2: Calculate the discriminant

The discriminant for a conic section is calculated as \(B^2 - 4AC\). For our equation, this becomes \(1^2 - 4(0)(0) = 1 - 0 = 1\).

Step 3: Determine the type of conic

A non-zero discriminant ( > 0) indicates a hyperbola, zero indicates a parabola, and less than zero indicates an ellipse. Here, since the discriminant is 1, it is a hyperbola.

Step 4: Use rotation to eliminate the \(xy\)-term

Since \(xy = 8\) is in the form \(xy = c\), this represents a hyperbola rotated \(45^\circ\). The rotation matrix is:\[\begin{pmatrix}\cos(45^\circ) & \sin(45^\circ) \-\sin(45^\circ) & \cos(45^\circ)\end{pmatrix}\]which simplifies the equation with the substitution \(X = \frac{x+y}{\sqrt{2}}\) and \(Y = \frac{-x+y}{\sqrt{2}}\). Therefore, \(XY = 8/2\) gives \(XY = 4\).

Step 5: Simplify the rotated equation

The equation \(XY = 4\) is a simpler form with no \(XY\)-term, ready for sketching. The coordinate system is aligned along the asymptotes of the hyperbola.

Step 6: Sketch the graph

Since \(XY = 4\) indicates a hyperbola with asymptotes along the lines \(X = 0\) and \(Y = 0\), it sketches as two branches in opposite quadrants around the origin. In the original \(xy\)-coordinate plane, these correspond to asymptotes along \(y = x\) and \(y = -x\).

Key Concepts

Discriminant

The discriminant is a helpful tool in determining the type of conic section represented by a given quadratic equation in two variables. For an equation in the standard form of a conic, the discriminant is calculated as \( B^2 - 4AC \). In our exercise with the equation given as \( xy = 8 \), we rearrange it to match the standard form, where the coefficients are \( A = 0, B = 1, C = 0 \). Calculating the discriminant, we find it to be \( 1^2 - 4 imes 0 imes 0 = 1 \).
A positive discriminant (greater than zero) indicates that the conic section is a hyperbola. If the discriminant is zero, the conic is a parabola. For a negative discriminant, it is an ellipse. In this case, since our discriminant is 1, we identify the equation as representing a hyperbola. This initial identification is crucial for guiding how we manipulate and eventually graph the equation.

Rotation of Axes

Rotation of axes is a technique used to eliminate the \(xy\)-term from the equation of a conic section. This makes the problem simpler, especially when sketching the graph. The presence of an \(xy\)-term indicates that the conic is rotated relative to the coordinate axes. For the equation \( xy = 8 \), the simplest rotation needed is by \(45^\circ\).
The transformation to a new set of axes \(X, Y\) is achieved using the rotation matrix:

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

These transformations align the axes with the asymptotes of the hyperbola. Substituting these into the original equation \( xy = 8 \) simplifies to \( XY = 4 \). This form no longer has an \(XY\)-term, making it easier to visualize and sketch. The rotation effectively straightens out the conic section into a more standard orientation.

Hyperbola

A hyperbola is one type of conic section that appears when slicing a cone with a plane in such a way that two opposite halves are formed, resembling an open curve with two branches. It is defined by the characteristic equation of the curve where the product of distances to two fixed points, called foci, is constant.
For the equation \( XY = 4 \), this describes a hyperbola in the \(XY\) plane. The hyperbola's asymptotes in this arrangement are along the lines \( X = 0 \) and \( Y = 0 \), representing the coordinate axes in the transformed system. In the original \(xy\)-coordinate system, these correspond to the lines \( y = x \) and \( y = -x \).
The significance of these asymptotes is that they are the lines that the branches of the hyperbola approach but never actually meet. The hyperbola's branches emerge symmetrically from around the center point at the origin, extending infinitely. When sketching the graph, these asymptotes provide a framework, ensuring the hyperbola opens up in the correct orientation.