Problem 13 - Precalculus: Mathematics for Calculus

Question

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 \sqrt{3} x y-y^{2}=4, \quad \phi=30^{\circ}$$

Step-by-Step Solution

Answer

The transformed conic equation is \[ \frac{3}{4}X^2 - \sqrt{3}XY + \frac{3}{4}Y^2 = 4 \].

Step 1: Convert Rotation Angle to Radians

To use the rotation formulas, we first convert the rotation angle \( \phi = 30^{\circ} \) to radians. We use the relation: \( 1^{\circ} = \frac{\pi}{180} \text{ radians} \). Therefore, \( 30^{\circ} = \frac{\pi}{6} \text{ radians} \).

Step 2: Identify Rotation Formulas

The general formulas for rotating the coordinates are: \[ x = X \cos\phi - Y \sin\phi \]\[ y = X \sin\phi + Y \cos\phi \]For \( \phi = 30^{\circ} = \frac{\pi}{6} \), \( \cos\phi = \frac{\sqrt{3}}{2} \) and \( \sin\phi = \frac{1}{2} \).

Step 3: Substitute Rotation Formulas Into Original Equation

Substitute the expressions for x and y in terms of X and Y into the original equation:\[ x^2 + 2\sqrt{3}xy - y^2 = 4 \]Substitute \( x = X\frac{\sqrt{3}}{2} - Y\frac{1}{2} \) and \( y = X\frac{1}{2} + Y\frac{\sqrt{3}}{2} \).

Step 4: Simplify the Expression

Compute each term by substitution:1. \(x^2 = \left(X \frac{\sqrt{3}}{2} - Y \frac{1}{2}\right)^2\)2. \(y^2 = \left(X \frac{1}{2} + Y \frac{\sqrt{3}}{2}\right)^2\)3. \(2\sqrt{3}xy = 2\sqrt{3}\left(X \frac{\sqrt{3}}{2} - Y \frac{1}{2}\right)\left(X \frac{1}{2} + Y \frac{\sqrt{3}}{2}\right)\)Combine and simplify these terms, expanding and collecting like terms to form an equation in terms of X and Y.

Step 5: Write the Simplified Equation

After expanding and simplifying, you will find the transformed equation in terms of X and Y:\[ \frac{3}{4}X^2 - \sqrt{3}XY + \frac{3}{4}Y^2 = 4 \]

Key Concepts

Coordinate Rotation

When we talk about coordinate rotation in the context of conic sections, we are referring to a transformation that rotates our axes by a specified angle. This is a crucial method since many conic sections have equations that become simpler when viewed from a different angle. During rotation, each point's coordinates are recalculated according to the angle of rotation.

Let's break it down:

• We start with a traditional coordinate system, and for the problem at hand, decide on an angle of rotation, such as 30°.
• Using rotation formulas, we convert each point in our system to align with this new, rotated set of axes. These formulas rely on trigonometric functions, specifically cosine and sine.
• The essence of the rotation formula lies in aligning the new axes such that our conic’s equation can be analyzed or simplified in another system.

In practice, by recalculating points using these formulas, we analyze the conic section through a fresh perspective, leading us often to a clearer understanding or a strategic simplification of equations.

Equation Transformation

Equation transformation is about rewriting the conic's equation in terms of new variables that represent the rotated coordinate system. This allows us to analyze the shape and properties of the conic from a new perspective or simply rearrange it into an easier form for solving or interpretation.

After determining our rotation angle and converting it to radians, we can express given equations of conics in terms of new variables, generally labeled as X and Y. This involves:

• Substituting our original variables with equations that include cosines and sines from the rotation angle.
• Example: original terms in x, y become complex expressions in terms of X, Y.
• Systematically simplify each part of the substituted equation to achieve the desired form.

Transformation often leads to an equation easier to work with or having a clearer geometric interpretation, as complex expressions in initial coordinates become simplified in the new coordinate form.

Radians Conversion

To work with rotations mathematically, especially in calculus and physics, we need to use radians rather than degrees. The reason is simple: radians are the standard unit of angular measure in mathematics, facilitating the application of trigonometric functions.

The conversion process from degrees to radians is straightforward:

• Recognize that a full circle is 360° or 2π radians.
• Thus, to convert degrees to radians, use the formula:
  
  \[ ext{radians} = ext{degrees} \times \frac{ ext{π}}{180} \]
• For example, a 30° angle, when converted, reads as \(30 \times \frac{π}{180} = \frac{π}{6}\) radians.

This conversion is fundamental for our rotation formulas and accurate in systems where mathematical precision is crucial, like conic section analysis.