Problem 19 - College Algebra

Question

For the following exercises, find a new representation of the given equation after rotating through the given angle. $$4 x^{2}-x y+4 y^{2}-2=0, \theta=45^{\circ}$$

Step-by-Step Solution

Answer

New equation: \(x'^2 + y'^2 = \frac{1}{2}\) (circle with radius \(\frac{1}{\sqrt{2}}\)).

Step 1: Identify the Transformation

We need to rotate the equation by an angle of \(\theta = 45^{\circ}\). The new coordinates \((x', y')\) after rotating can be represented using the equations:\[x = x' \cos \theta - y' \sin \theta\]\[y = x' \sin \theta + y' \cos \theta\]For \(\theta = 45^{\circ}\), the transformations become:\[x = \frac{1}{\sqrt{2}}(x' - y')\]\[y = \frac{1}{\sqrt{2}}(x' + y')\]

Step 2: Substitute for Rotation

Substitute \(x = \frac{1}{\sqrt{2}}(x' - y')\) and \(y = \frac{1}{\sqrt{2}}(x' + y')\) into the original equation:\[4x^2 - xy + 4y^2 - 2 = 0\]This will give:\[4\left(\frac{1}{\sqrt{2}}(x' - y')\right)^2 - \left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right) + 4\left(\frac{1}{\sqrt{2}}(x' + y')\right)^2 - 2 = 0\]

Step 3: Expand and Simplify

Expand each term separately:1. \(4\left(\frac{1}{\sqrt{2}}(x' - y')\right)^2 = 2(x' - y')^2\)2. \(\left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right) = \frac{1}{2}(x'^2 - y'^2)\)3. \(4\left(\frac{1}{\sqrt{2}}(x' + y')\right)^2 = 2(x' + y')^2\)Substitute these back into the transformed equation to further simplify.

Step 4: Combine Like Terms

Combining the expanded terms from the previous step, we have: \[2(x'^2 - 2x'y' + y'^2) - \frac{1}{2}(x'^2 - y'^2) + 2(x'^2 + 2x'y' + y'^2) - 2 = 0\]Simplify the combined expression:\[4x'^2 + 4y'^2 - 2 = 0\]

Step 5: Final Simplified Equation

The final equation represents the new rotated form of the original equation.\[2x'^2 + 2y'^2 - 1 = 0\]This can be written as:\[x'^2 + y'^2 = \frac{1}{2}\]This is the equation of a circle centered at the origin with radius \(\frac{1}{\sqrt{2}}\).

Key Concepts

Angle Transformation

In coordinate geometry, rotating a plane about the origin by a given angle is a common technique used to simplify equations. Here, the angle transformation helps us change our coordinate system without altering the inherent properties of the geometric shape. To rotate by an angle \( \theta \), we can use the formulas:

• \( x = x' \cos \theta - y' \sin \theta \)
• \( y = x' \sin \theta + y' \cos \theta \)

For \( \theta = 45^{\circ} \), these formulas simplify using the known trigonometric values \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{1}{\sqrt{2}} \). Substituting these values, we obtain:

• \( x = \frac{1}{\sqrt{2}}(x' - y') \)
• \( y = \frac{1}{\sqrt{2}}(x' + y') \)

This transformation is pivotal as it allows us to substitute these expressions into the original equation, leading to a more manageable form. This ultimately reveals more about the geometric nature of the equation, such as its symmetry and shape.

Equation Simplification

After making substitutions into the original equation for rotated coordinates, a crucial step involves simplification. This process entails expanding and reworking the algebraic expressions to combine like terms and reduce complexity. Initially, we substitute \( x = \frac{1}{\sqrt{2}}(x' - y') \) and \( y = \frac{1}{\sqrt{2}}(x' + y') \) into the equation:\[ 4x^2 - xy + 4y^2 - 2 = 0 \]Replacing accordingly gives us a transformed equation with more terms. The task at hand is to carefully expand each term:

• \( 4\left(\frac{1}{\sqrt{2}}(x' - y')\right)^2 = 2(x' - y')^2 \)
• \( \left(\frac{1}{\sqrt{2}}(x' - y')\right)\left(\frac{1}{\sqrt{2}}(x' + y')\right) = \frac{1}{2}(x'^2 - y'^2) \)
• \( 4\left(\frac{1}{\sqrt{2}}(x' + y')\right)^2 = 2(x' + y')^2 \)

By replacing these back, we proceed to combine like terms, tackling each segment one at a time. The result reveals a streamlined expression. Simplifications like this are essential as they untangle complex equations into more straightforward ones that are easy to interpret and solve.

Circle Equation

Upon final simplification of the equation, it becomes evident that the new form is that of a circle. Previously, the equation was more convoluted, but after transformation and reduction, it beautifully simplifies into:\[ x'^2 + y'^2 = \frac{1}{2} \]This equation represents a circle centered at the origin \((0,0)\) with a radius given by \( \frac{1}{\sqrt{2}} \). Recognizing this as a circle is crucial in understanding how equations represent different geometric shapes based on their forms. The simplicity of this equation speaks to the power of coordinate rotation and simplification, offering a clear picture of the geometric figure in question. Understanding how to identify and work with the circle equation in both standard forms and transformed forms enhances problem-solving techniques in geometry and algebra.

Trigonometric Identities

Trigonometric identities play a vital role in coordinate rotation, especially when dealing with angles like \( 45^{\circ} \). Fundamental identities such as \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{1}{\sqrt{2}} \) allow us to transform equations accurately without altering their geometric nature. These identities help maintain equal sides and angles by ensuring the transformation remains consistent and precise.When examining problems involving rotation or transformation, being familiar with common trigonometric identities is invaluable. They can greatly simplify the substitution process as seen here, turning potentially daunting expressions into something manageable. Additionally, they support the translation from complex trigonometric operations back into basic algebraic forms, facilitating a clearer understanding of the underlying geometric relationships. This deepens one’s ability to maneuver effectively through geometry and trigonometric problems.