Problem 24 - Contemporary Precalculus

Question

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 x^{2}+2 \sqrt{2} x y+2 y^{2}-12=0$$

Step-by-Step Solution

Answer

Based on the given equation, compute the discriminant and determine the type of conic section it represents. After identifying it as an ellipse, find an appropriate viewing window to display the complete graph. The discriminant is -16, meaning the equation represents an ellipse. The viewing window can be estimated using graphing software, with the x-axis ranging from -5 to 5 and the y-axis ranging from -3 to 3.

Step 1: Compute the discriminant

Given the equation of a general conic section: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ we can compute the discriminant as follows: $$\Delta = B^2 - 4AC$$ For the given equation, we have: - A = 3 - B = 2√2 - C = 2 Now, let's compute the discriminant: $$\Delta = (2\sqrt{2})^2 - 4(3)(2)$$

Step 2: Determine the type of conic section

Now that the discriminant has been computed, we can determine the type of conic section it represents: - If \(\Delta > 0\), the equation represents a hyperbola. - If \(\Delta = 0\), the equation represents a parabola. - If \(\Delta < 0\), the equation represents an ellipse. Computing the discriminant: $$\Delta = (2\sqrt{2})^2 - 4(3)(2) = 8 - 24 = -16$$ Since the discriminant is less than 0 (\(\Delta < 0\)), the given equation represents an ellipse.

Step 3: Find the viewing window

Since we know the equation represents an ellipse, we can find the viewing window that shows the complete graph. We'll need to find its semi-major axis, semi-minor axis, center, and rotation angle. To find these, transform the given ellipse equation to its standard form of: $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$ Here, \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. To do this, we can use a method called principal axis decomposition or linear algebraic methods. These procedures exceed the scope of high school math. However, a viewing window can be found using trial and error or graphing software. For this ellipse, we can estimate a viewing window with the help of a graphing calculator or software like MATLAB, Mathematica, or Desmos, or by doing trial and error. After trying some suitable values for x and y, we could find the viewing window as: - x-axis: from -5 to 5 - y-axis: from -3 to 3 This viewing window would show the complete graph of the ellipse.

Key Concepts

Discriminant

The discriminant is a vital tool in determining the type of conic section represented by a specific equation. It gives us insight into the geometry of the curve described. The general equation for a conic section is given by:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]For this equation, the discriminant \( \Delta \) is calculated using the formula:\[\Delta = B^2 - 4AC\]The discriminant tells us:

• If \(\Delta > 0\), the equation represents a hyperbola.
• If \(\Delta = 0\), the equation represents a parabola.
• If \(\Delta < 0\), the equation represents an ellipse.

In the given exercise, we calculated the discriminant using the coefficients \(A = 3\), \(B = 2\sqrt{2}\), and \(C = 2\). By substituting these into the formula, we found that \(\Delta = -16\). Since this value is less than zero, the equation represents an ellipse.

Ellipse

Ellipses are fascinating geometric figures that can be easily identified once the discriminant indicates \(\Delta < 0\). An ellipse in the standard form has an equation of:\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]Here:

• \((h, k)\) is the center of the ellipse.
• \(a\) is the semi-major axis, the longest diameter of the ellipse.
• \(b\) is the semi-minor axis, the shortest diameter of the ellipse.

The axes of an ellipse help in determining its orientation and size. While converting to this standard form involves complex algebraic manipulations, which may exceed basic high school algebra, understanding these geometric parameters is crucial. With these parameters, we can quickly grasp the shape and size of the ellipse we're dealing with.

Viewing Window

The concept of a viewing window is invaluable when graphing conic sections like ellipses, especially when using digital graphing tools or calculators. A viewing window determines the section of the coordinate plane you will view or graph, ensuring the entire curve is displayed accurately. For an ellipse, selecting an appropriate viewing window means ensuring both the semi-major and semi-minor axes are fully visible. This involves trial and error or using software to pinpoint the correct ranges. In our example, a suitable viewing window was estimated to be:

• Along the x-axis, from -5 to 5.
• Along the y-axis, from -3 to 3.

This range ensures that the complete ellipse is visible, allowing for easier analysis and comprehension. Graphing software such as Desmos, MATLAB, or a graphing calculator often help find these window settings more precisely and efficiently.