Problem 9 - 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. $$x^{2}-3 y^{2}=4, \quad \phi=60^{\circ}$$

Step-by-Step Solution

Answer

The equation in new coordinates is \(X^2 - 3Y^2 = 4\).

Step 1: Understand the Rotation Angle

The problem states that the coordinate axes are rotated by an angle of \( \phi = 60^{\circ} \). This means we need to find new coordinates (\(X, Y\)) using the formulas for rotation of axes. Specifically, \( X = x \cos \phi + y \sin \phi \) and \( Y = -x \sin \phi + y \cos \phi \).

Step 2: Substitute Rotation Formulas

Substitute \( \phi = 60^{\circ} \), \( \cos 60^{\circ} = \frac{1}{2} \), and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) into the rotation formulas to get:\[X = \frac{1}{2}x + \frac{\sqrt{3}}{2}y\]\[Y = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y\]

Step 3: Express Original Coordinates in Terms of New Coordinates

Solve these equations for \( x \) and \( y \) in terms of \( X \) and \( Y \). You have:\[x = \frac{1}{2}X - \frac{\sqrt{3}}{2}Y\]\[y = \frac{\sqrt{3}}{2}X + \frac{1}{2}Y\]

Step 4: Substitute Back into Original Equation

Substitute \( x \) and \( y \) into the original equation \( x^2 - 3y^2 = 4 \):\[\left(\frac{1}{2}X - \frac{\sqrt{3}}{2}Y\right)^2 - 3\left(\frac{\sqrt{3}}{2}X + \frac{1}{2}Y\right)^2 = 4\]

Step 5: Expand and Simplify

Expand each square and simplify the expression.- Expansion:\[= \left(\frac{1}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2\right)\]\[- \left(3\left(\frac{3}{4}X^2 + \frac{\sqrt{3}}{2}XY + \frac{1}{4}Y^2\right)\right)\]Combine the terms to simplify:\[= X^2 - 3Y^2\]Thus, the equation of the conic in the rotated XY-coordinates is:\[X^2 - 3Y^2 = 4\]

Step 6: Conclusion

The equation of the rotated conic in the new XY-coordinates is \( X^2 - 3Y^2 = 4 \). This shows that the rotation does not change the form of the equation, as the angle used aligns with the symmetry of the conic.

Key Concepts

Rotation of Axes

When analyzing conic sections, sometimes it becomes necessary to rotate the coordinate axes. This is typically done to simplify the equation of the conic. For instance, in this problem, the original equation is given as \(x^2 - 3y^2 = 4\), and we are asked to rotate the axes by \(60^{\circ}\). By utilizing the rotation of axes formulas, we can convert the old coordinates \((x, y)\) to new ones \((X, Y)\).

The transformation formulas used here are:

• \(X = x \cos \phi + y \sin \phi\)
• \(Y = -x \sin \phi + y \cos \phi\)

Where \(\phi\) is the angle of rotation. This results in the new coordinates \(X\) and \(Y\) that align differently, often simplifying the description of the conic. The process involves understanding the trigonometrical components like sine and cosine of the angle to establish these connections.

By using these formulas, any complex alignment of conics can be adjusted, allowing easier calculation and classification.

Coordinate Transformation

The step of coordinate transformation involves substituting the rotation formulas into the original conic equation. This procedure transforms the problem's context, allowing us to see the structure more clearly through the new coordinates.

After determining the angle of rotation, replace \(\phi\) with its value, \(60^{\circ}\), compute \(\cos 60^{\circ}\) and \(\sin 60^{\circ}\), which equal \(\frac{1}{2}\) and \(\frac{\sqrt{3}}{2}\) respectively.
Then apply these into the rotation formulas:

• \(X = \frac{1}{2}x + \frac{\sqrt{3}}{2}y\)
• \(Y = -\frac{\sqrt{3}}{2}x + \frac{1}{2}y\)

Next, it's vital to express \(x\) and \(y\) in terms of \(X\) and \(Y\), which involves reversing these equations.

• \(x = \frac{1}{2}X - \frac{\sqrt{3}}{2}Y\)
• \(y = \frac{\sqrt{3}}{2}X + \frac{1}{2}Y\)

By substituting these values back into the original conic equation, you can transform the entire equation to lie within the new coordinate framework, revealing any structural symmetries hidden in the old system.

Equation of Conic

The result of applying rotation and coordinate transformations to a conic's equation is to present an easier form for understanding or further calculations. In our example, after inputting the transformed coordinates into the original equation \(x^2 - 3y^2 = 4\), and simplifying, we find that the equation remains structurally unchanged as \(X^2 - 3Y^2 = 4\).

First, expand the transformed terms separately:

• For \(\left(\frac{1}{2}X - \frac{\sqrt{3}}{2}Y\right)^2\)
• For \(3\left(\frac{\sqrt{3}}{2}X + \frac{1}{2}Y\right)^2\)

Combine the expansions:
- Calculated individually first, then subtracted and grouped:
- \(\left(\frac{1}{4}X^2 - \frac{\sqrt{3}}{2}XY + \frac{3}{4}Y^2\right) - 3\left(\frac{3}{4}X^2 + \frac{\sqrt{3}}{2}XY + \frac{1}{4}Y^2\right)\)
Upon complete expansion and simplification, recognize the given equation in its transformed state. The fact that it remains in an equivalent form like \(X^2 - 3Y^2 = 4\) points to stability of property within symmetrical alignment, often a noteworthy aspect of conic equations under axis transformations.

Recognizing this final form helps quickly categorize the conic based on symmetry properties endowed by the rotation.