Problem 22 - Precalculus: Mathematical 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 \(x y\) -term. (c) Sketch the graph. $$153 x^{2}+192 x y+97 y^{2}=225$$

Step-by-Step Solution

Answer

The equation represents an ellipse after eliminating the xy-term using rotation.

Step 1: Identify the Discriminant

The discriminant for a conic section given by the general equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) is calculated using the formula \(B^2 - 4AC\). In this equation, \(A = 153\), \(B = 192\), and \(C = 97\). Substitute these values into the formula to get the discriminant: \(192^2 - 4 \times 153 \times 97\).

Step 2: Calculate the Discriminant

Compute the value of the discriminant \(192^2 - 4 \times 153 \times 97\). We find \(192^2 = 36864\) and \(4 \times 153 \times 97 = 59364\). Thus, the discriminant is \(36864 - 59364 = -22500\).

Step 3: Classify the Conic Section

Based on the discriminant value, if it is less than zero, the conic is an ellipse. Since the discriminant calculated is \(-22500\), which is less than zero, the equation represents an ellipse.

Step 4: Rotation of Axes Formulas

Use the formulas for rotating axes \(x = x' \cos \theta - y' \sin \theta\) and \(y = x' \sin \theta + y' \cos \theta\) with \(\tan 2\theta = \frac{B}{A-C}\) to eliminate the \(xy\)-term. Here, \(\tan 2\theta = \frac{192}{153-97} = \frac{192}{56}\).

Step 5: Determine Rotation Angle

Calculate \(\theta\) using \(\tan 2\theta = \frac{192}{56}\). Compute \(\theta\) by finding \(\tan \theta\), through the half-angle identity, and solve to get \(\theta\). This eliminates the \(xy\)-term in the new coordinates \(x', y'\).

Step 6: Substitute into the Equation

Substitute \(x'\) and \(y'\) back into the original equation in place of \(x\) and \(y\) and simplify. The resulting equation will have only \(x'^2\) and \(y'^2\) terms, confirming the ellipse.

Step 7: Sketch the Graph

In the \(x'y'\)-coordinate system, sketch an ellipse centered at the origin with semi-major and semi-minor axes along the \(x'\) and \(y'\) directions, respectively. The lengths of these axes depend on the coefficients found in the simplified equation.

Key Concepts

Understanding the Discriminant in Conic Sections

The discriminant is a vital tool in examining conic sections. In the general form of a conic equation, \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is given by the formula \( B^2 - 4AC \). This number helps us classify the type of conic we are dealing with.

• If the discriminant is greater than zero, the conic is a hyperbola.
• If it equals zero, the conic is a parabola.
• A discriminant less than zero indicates an ellipse or a circle, as circles are special cases of ellipses.

In our exercise, calculating the discriminant resulted in \(-22500\), which tells us that our equation represents an ellipse.

Why does this matter? Each type of conic has distinct properties and shapes. Knowing that we have an ellipse helps us focus on the features that are crucial for ellipses, like their axes and centers.

The Magic of Rotation of Axes

Conic sections can sometimes contain an \(xy\)-term, which complicates their graphs. To handle this, we employ rotation of axes to remove it. This process involves rotating the coordinate system by an angle \(\theta\), calculated using the formula \(\tan 2\theta = \frac{B}{A - C} \).

For example, in the equation \(153x^2 + 192xy + 97y^2 = 225\), we found \(\tan 2\theta\) as \(\frac{192}{56}\). Solving for \(\theta\) changes the coordinates \((x, y)\) to \((x', y')\) with new axes. The rotation effectively removes the \(xy\)-term, leading to simpler and more recognizable equations.

This technique is like turning a picture to avoid seeing it at a strange angle, letting us see its true form. By removing the cross-product term, we can more easily categorize and ultimately sketch the graph.

Recognizing and Sketching Ellipses

Ellipses are elegant shapes defined by their symmetrical features around two axes: the major and minor axes. They appear when the discriminant of a conic section equation is negative.

Once identified, ellipses can be described using their standard form equation:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

When sketching an ellipse, it’s essential to locate its center and the lengths of its axes:

• The center provides the starting point for the ellipse.
• The axes indicate the extent of the ellipse in both x and y directions.

After rotating the axes and rewriting\((x, y)\) in terms of \((x', y')\), you can visually lay out the ellipse, ensuring it fits symmetrically along its new axis, thus offering a clearer map of its shape.