Problem 21 - Precalculus

Question

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$x^{2}+2 x y+y^{2}+\sqrt{2} x-\sqrt{2} y=0$$

Step-by-Step Solution

Answer

After rotating the axes by 45 degrees, the equation takes the form: \(x'^2 + y'^2 = 0\), which represents a point at the origin. This is then plotted onto a graph, showing both original and rotated axes.

Step 1: Identify the Angle of Rotation

For the given equation, the coefficient before the xy term is 2. To get rid of this term, one must rotate the coordinate system to create a new one where this term disappears. The angle of rotation can be found using the following formula for this type of quadratic equations: \(\theta = 0.5 \arctan (2/0)\). Since the denominator is zero, the angle of rotation can't be determined directly and we assume it as \(\theta = 45^\circ\).

Step 2: Apply the Rotation Transformation

The transformation of coordinates from the old (x, y) to the new (x', y') system is done by using the rotation matrix equations: \( x= x' \cos(\theta) - y' \sin(\theta) \) and \(y= x' \sin(\theta) + y' \cos(\theta) \). These equations will replace x and y in the original equation.

Step 3: Substitute the Transformed Coordinates into the Equation

Substitute the rotated coordinates from step 2 into the original equation in place of x and y. The transformed equation will become: \((x'\cos(\theta) - y'\sin(\theta))^{2}+2(x'\cos(\theta) - y'\sin(\theta))(x'\sin(\theta) + y'\cos(\theta))+(x'\sin(\theta) + y'\cos(\theta))^{2}-\sqrt{2}(x'\cos(\theta) - y'\sin(\theta))+\sqrt{2}(x'\sin(\theta) + y'\cos(\theta))=0\). Since \(\theta = 45^\circ\), both \(\cos(45^\circ)\) and \(\sin(45^\circ)\) are equal to \(\frac{\sqrt{2}}{2}\). After inserting these values and simplifying the equation, the xy term will disappear.

Step 4: Plot the Graph

Plot the graph of the transformed equation showing both the original (x, y) and new (x', y') sets of axes. The plot provides better understanding of how the rotation transformation affected the original figure.

Key Concepts

Coordinate Rotation

Coordinate rotation is a mathematical technique used to simplify equations by rotating the coordinate axes. This often helps to eliminate confusing mixed terms like the \(xy\)-term in quadratic equations.
Imagine the original x-y axes being tilted around a point. Such a rotation transforms the coordinates into a new coordinate system, denoted as \(x'\) and \(y'\).
The angle of rotation, \(\theta\), dictates the degree of this rotation. For bilinear terms like \(2xy\), the goal is to find \(\theta\) that eliminates the term.

• Use the formula \(\theta = 0.5 \arctan (\frac{B}{A-C})\), derived from the general form of a conic.
• In our exercise, given that the denominator becomes zero, we assume \(\theta = 45^\circ\).

Next, we use rotation equations:
\[x = x'\cos(\theta) - y'\sin(\theta)\] \[ y = x'\sin(\theta) + y'\cos(\theta)\]
This allows us to substitute \(x\) and \(y\) in our equation with \(x'\) and \(y'\), resulting in the new, simplified equation.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They have the general form of \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
This form can have an \(xy\)-term, referred to as the off-diagonal term, which can complicate solving or visualizing the equation.

• The coefficients \(A\), \(B\), and \(C\) represent the quadratic parts of the equation.
• Rotating the coordinate system often aims to eliminate \(B\), simplifying the equation.

The simplified form can further be manipulated or graphed with ease.
Removing the \(xy\)-term leads us to a rotation of coordinates, where the quadratic part becomes easier to factor or sketch.

Standard Form Conversion

Standard form conversion helps in transforming the quadratic equation to a more recognizable and easily usable form.
After performing a coordinate rotation, a complicated quadratic equation can be simplified into its standard form, \((x')^2/a^2 + (y')^2/b^2 = 1\) for ellipses, for example.

• The rotated equation loses the \(xy\)-term, resuming to a more standard quadratic equation.
• This makes it easier to recognize and plot as a conic section, like a circle or ellipse, based on the coefficients after rotation.

The standard form provides insight into the graph's characteristics such as position, size, and orientation concerning the new axes \(x'\) and \(y'\).
This is crucial for graphical analysis and understanding the geometric nature of the quadratic relation.

Graph Sketching

Once the equation is in its standard form, sketching its graph becomes intuitive.

• Graph sketching involves plotting the shape defined by your simplified equation.
• This visual representation helps in fully understanding the effect of the coordinate rotation.

In the original task, after adjusting to eliminate the \(xy\)-term, the graph showcases a figure symmetric to the new coordinate axes \(x'\) and \(y'\).
Plot both the original and the new rotated axes on the graph.
This allows one to see explicitly how transforming the equation has affected the orientation and shape of the graph
Having both axes depicted helps you see how the original scene changes through geometric transformations.