Question

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. 4 x^{2}+x y+4 y^{2}=56

Short answer

After rotating by \( 45^\circ \), the curve is \( x'^2 + y'^2 = \frac{56}{5} \) (a circle).

Step-by-step solution

Identify the Cross-Product Term

The given equation is \(4x^2 + xy + 4y^2 = 56\). The term \(xy\) is the cross-product term that we need to eliminate by rotating the axes.

Calculate Rotation Angle for Zero Cross-Product Term

To eliminate the cross-product term, we use the rotation formulas. Using \( B = 1 \) for the \( xy \) term, and \( A = 4 \), \( C = 4 \), the angle \( \theta \) for rotation is given by \( \tan 2\theta = \frac{B}{A-C} = \frac{1}{0} \), which means \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \). So, \( \theta \) is \( 45^\circ \).

Apply Rotation Formulas

With \( \theta = 45^\circ \), use the rotation formulas: \( x = x'\cos\theta - y'\sin\theta \) and \( y = x'\sin\theta + y'\cos\theta \). Substitute \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \). Transforming gives: \( x = \frac{\sqrt{2}}{2}(x' - y') \) and \( y = \frac{\sqrt{2}}{2}(x' + y') \).

Substitute and Simplify the Equation

Substitute \( x = \frac{\sqrt{2}}{2}(x' - y') \) and \( y = \frac{\sqrt{2}}{2}(x' + y') \) into \( 4x^2 + xy + 4y^2 = 56 \). Simplifying, the equation eliminates the cross-product term, resulting in \( 5(x'^2 + y'^2) = 56 \) after simplification.

Convert to Standard Form by Completing the Square

The simplified form \( 5(x'^2 + y'^2) = 56 \) is already in a form that represents a circle. Divide through by 5 to get \( x'^2 + y'^2 = \frac{56}{5} \), indicating a circle centered at the origin (0, 0) with radius \( \sqrt{\frac{56}{5}} \). No further translation is needed.

Graph the Equation with Rotated Axes

Plot the circle with center (0, 0) and radius \( \sqrt{\frac{56}{5}} \). Include the rotated axes which were previously expressed from the original coordinates \( x \) and \( y \) using the rotation angle \(45^\circ\). The axes have shifted by the calculated rotation.

Key concepts

Cross-Product Term

In the context of quadratic equations, the cross-product term is the term that includes the product of two different variables, typically in the form of \(xy\). The presence of this term complicates the equation and often indicates that the graph is not aligned with the coordinate axes. To eliminate this term, we use a method known as rotation of axes. This involves rotating the entire coordinate system by a certain angle so that the new axes align with the graph's natural axes, thus eliminating the cross-product term. For the given equation \(4x^2 + xy + 4y^2 = 56\), the cross-product term is \(xy\). To find the angle of rotation, we use the relationship \( \tan 2\theta = \frac{B}{A-C} \), where \(A\), \(B\), and \(C\) are coefficients of the quadratic terms. This formula helps in determining the correct angle to achieve a new coordinate system where the cross-product term vanishes, making further analysis simpler.

Standard Form

Converting a conic section equation to standard form involves eliminating the cross-product term and simplifying the equation such that it fits into one of the standard conic forms: circles, ellipses, hyperbolas, or parabolas. In our example, after eliminating the cross-product term by rotating the axes, the equation simplifies to \(5(x'^2 + y'^2) = 56\). This equation represents a circle in the standard form \((x' - h)^2 + (y' - k)^2 = r^2 \) centered at the origin \((h, k) = (0, 0)\) and having a radius \(r\). Further simplification by dividing every term by 5 converts it to \(x'^2 + y'^2 = \frac{56}{5}\). The standard form reveals the fundamental characteristics of the shape, such as its center and radius, which tells us it's a circle.

Complete the Square

Completing the square is a method used in algebra to transform a quadratic equation into a perfect square trinomial form. This technique is helpful for both solving quadratic equations and for converting certain conic sections into their standard form. However, in this specific problem, completing the square was not necessary after rotation, because the equation \( 5(x'^2 + y'^2) = 56 \) already represented a complete square, indicating a circle. For equations warranting this step, the goal is to manipulate the equation into a form like \((x' - h)^2 + (y' - k)^2 = r^2\), which clearly defines the graph. This is frequently needed when the equation after rotation still has linear terms or doesn’t immediately suggest which standard conic form it fits.

Graphing

Graphing a rotated equation involves plotting it with respect to the new rotated axes calculated after eliminating the cross-product term. For the transformed equation \(x'^2 + y'^2 = \frac{56}{5}\), the graph is a circle with its center at the origin of the rotated axes \((x', y')\). The radius of this circle is \(\sqrt{\frac{56}{5}}\). While plotting, it's essential to visualize how the axes have shifted by a \(45^\circ\) rotation, making \(x'\) and \(y'\) the new horizontal and vertical axes. This helps in capturing the geometric properties correctly in relation to the overall coordinate plane. Particularly in examinations or homework, drawing the original axes and the rotated axes can provide a clearer understanding of how the transformation affects the shape of the graph, reinforcing the geometric intuition behind rotation of axes.