Problem 48 - Algebra 2

Question

An \(x y\) term has an interesting effect on the graph of a conic section. Sketch the graph of each conic section below using your graphing calculator. (Hint: To solve for \(y,\) you will need to complete a square.) a. \(4 x^{2}+2 x y+y^{2}=9\) b. \(4 x^{2}+2 x y-y^{2}=9\)

Step-by-Step Solution

Answer

The first equation represents a rotated ellipse with major axis 3 units and minor axis 1.5 units. The second equation represents a rotated hyperbola.

Step 1: Identify and Rewite the equations

Take the first equation \(4x^2 + 2xy + y^2 = 9\). This is a format of a rotated ellipse. We rewrite the given equation in its standard format by completing the square.

Step 2: Calculate the co-efficients for the standard format - Part I

We first divide through by 9 (since a rotated ellipse has a standard form of \(x'^2/a^2 + y'^2/b^2 = 1\)): \(4/9x^2 + 2/9xy + 1/9y^2 = 1\). Then, we equate this to the standard format equation of a rotated ellipse: \(x'^2/a^2 + y'^2/b^2 = 1\), where \(x' = x\cosθ - y\sinθ\), \(y' = x\sinθ + y\cosθ\), \(a = \sqrt{b^2/(1 - (b^2/a^2))}\) and \(b = \sqrt{a^2/(1 - (a^2/b^2))}\). By comparing the coefficients on both sides, we find that \(a^2 = 9/4\), and \(b^2 = 9\). Thus, \(a = 3/2\) and \(b = 3\). Therefore, the standard form of our ellipse is \(x'^2/2.25 + y'^2/9 = 1\).

Step 3: Rotate and Graph the Conic Section - Part I

Now we have these information, the equation of ellipse can be graphed. Since this form of the ellipse equation is a rotation of the original ellipse equation, the graph will be a rotated ellipse with major axis 3 units and minor axis 1.5 units.

Step 4: Identify and Rewrite the equation - Part II

Now, take the second equation \(4x^2 + 2xy - y^2 = 9\). This is again a format of a rotated ellipse. We attempt to rewrite this equation in its standard format by rearranging the terms.

Step 5: Calculate the co-efficients for the standard format - Part II

We divide through by 9: \(4/9x^2 + 2/9xy - 1/9y^2 = 1\). Equating this to the standard format equation of a rotated hyperbola \(x'^2/a^2 - y'^2/b^2 = 1\), we find that \(a^2 = 9/4\) and \(b^2 = 9\), which gives \(a = 3/2\) and \(b = 3\). Therefore, the standard format of our hyperbola is \(x'^2/2.25 - y'^2/9 = 1\).

Step 6: Rotate and Graph the Conic Section - Part II

With this information, graph the hyperbola. Since this form of the hyperbola equation is a rotation of the original one, the graph will be a rotated hyperbola.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that surrounds two focal points. The sum of the distances from any point on the ellipse to the two foci is constant. In mathematical terms, the general equation for an ellipse in a standard form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center of the ellipse, and \(a\) and \(b\) represent the horizontal and vertical distances from the center to the ellipse's edge respectively.

• When \(a > b\), the ellipse is elongated along the x-axis (horizontal axis).
• Conversely, when \(b > a\), the ellipse stretches along the y-axis (vertical axis).

Ellipses are important in mathematics and physics; for instance, they describe planetary orbits in astronomy as an effect of gravity. Understanding ellipses includes recognizing rotations, which occur when the axes of symmetry are not parallel to the coordinate axes. This task involves completing the square to convert the general form into the standard form.

Hyperbola

Hyperbolas are another type of conic section, distinguished by the difference in squares in their equations. The standard form for a hyperbola is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) (or vice versa), representing two separate curves called branches.

• The transverse axis, defined by \(a\), connects the vertices of the hyperbola.
• The conjugate axis, defined by \(b\), is perpendicular to the transverse axis.

Each branch of the hyperbola approaches two lines called asymptotes, which intersect at the center of the hyperbola \((h, k)\). Hyperbolas appear in real-life scenarios like radio waves and hyperbolic trajectories in physics. Similar to ellipses, hyperbolas can also require completing the square to rewrite and rotate their equations. In the exercise above, recognizing the equation as a hyperbola is essential to transforming it into its proper graphable form.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a more useful form. It is particularly handy when dealing with conic sections like ellipses and hyperbolas to uncover hidden transformations like rotations.
The general form \(ax^2 + bxy + cy^2 = k\) is not immediately useful for graphing. Completing the square takes terms \(ax^2\) and \(cy^2\) and forms a square by adding and subtracting necessary amounts.
For example, to complete the square in one variable, split the linear coefficient, take half of it, square it, and add it to both sides of the equation:

• Consider \(x^2 + bx\). Take \((b/2)^2\) and add \((b/2)^2 - (b/2)^2\).
• This results in \((x + b/2)^2\), simplifying the equation to a more manageable form.

This approach helps rewrite and solve the conic sections, allowing them to be plotted accurately. In the given exercise, completing the square transforms complex quadratic equations into ellipses and hyperbolas, thus aiding in visualizing the graphs.