Problem 13 - Precalculus Mathematics for Calculus

Question

Determine the equation of the given conic in \(X Y\)-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 equation in rotated coordinates is \(X^2 - Y^2 = \frac{16}{5}\).

Step 1: Identify Rotation Angle and Coefficients

The given angle of rotation is \(\phi = 30^{\circ}\). The given equation of the conic is \(x^2 + 2\sqrt{3}xy - y^2 = 4\). We need to identify the coefficients: \(A = 1\), \(B = 2\sqrt{3}\), and \(C = -1\). These coefficients will be used in our transformation formulae.

Step 2: Calculate Rotation Formula Components

The formulas for transforming coordinates involve trigonometric functions of \(\phi\). We have:- \(\cos(\phi) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)- \(\sin(\phi) = \sin(30^{\circ}) = \frac{1}{2}\). Now we can use these values during transformation.

Step 3: Calculate New Coefficients in Rotated System

Using the trigonometric values found in step 2, calculate the new coefficients using:\[A' = A\cos^2(\phi) + B\cos(\phi)\sin(\phi) + C\sin^2(\phi),\]\[B' = 2(A - C)\cos(\phi)\sin(\phi) + B(\cos^2(\phi) - \sin^2(\phi)),\]\[C' = A\sin^2(\phi) - B\cos(\phi)\sin(\phi) + C\cos^2(\phi).\]Substituting in the values:\[A' = 1\left(\frac{3}{4}\right) + 2\sqrt{3}\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right) - 1\left(\frac{1}{4}\right) = \frac{5}{4}\]\[B' = 2(1)\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right) + 2\sqrt{3}\left(\frac{3}{4} - \frac{1}{4}\right) = 0\]\[C' = 1\left(\frac{1}{4}\right) - 2\sqrt{3}\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right) - 1\left(\frac{3}{4}\right) = -\frac{5}{4}\].

Step 4: Write the Equation in New Coordinates

Using the values calculated in Step 3, the equation of the conic in the new \(XY\)-coordinate system is of the form:\[A'X^2 + B'XY + C'Y^2 = D,\] where \(D = 4\). Since \(B' = 0\), substitute to get:\[\frac{5}{4}X^2 - \frac{5}{4}Y^2 = 4.\] Dividing through by \(\frac{5}{4}\), the equation simplifies to:\[X^2 - Y^2 = \frac{16}{5}.\]

Step 5: Conclusion

The new equation, after rotation by \(30^{\circ}\), represents an unchanged form but in transformed coordinates. It maintains its structure as \(X^2 - Y^2 = \frac{16}{5}\), signaling a hyperbola.

Key Concepts

Coordinate Rotation

When we need to analyze conic sections, such as ellipses, parabolas, or hyperbolas, sometimes the shape appears skewed because of an oblique orientation. To simplify these equations and gain insights easily, we rotate the coordinate system. Coordinate rotation involves transforming the original axes to new, rotated ones. This transformation is especially useful to eliminate the cross-product term,

• When performing coordinate rotation, we aim to align the new X and Y axes in such a way that any linear terms in the product of variables, such as the XY term, are removed or simplified.
• The rotation angle, \(\phi\), indicates by how much and in which direction (clockwise or counterclockwise) the rotation should occur.

To achieve this rotation mathematically, specific trigonometric identities are employed, dependent on \(\cos(\phi)\) and \(\sin(\phi)\). By applying these transformations, conic equations are re-expressed in a more straightforward form, which makes it easier to identify and study their geometric properties.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying calculations during coordinate rotation. They allow us to use known angles in formulas to transform coordinates effectively:

• One fundamental identity is \(\cos^2(\phi) + \sin^2(\phi) = 1\), reflecting the Pythagorean theorem.
• For a specific example like \(\phi = 30^\circ\), it helps to know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(30^\circ) = \frac{1}{2}\).
• These values are used as components in the transformation equations, affecting the coefficients of our new, rotated conic equation.

Trigonometric identities are not just about sine and cosine. By understanding their relationships, you simplify the equations and make complex geometrical properties more visible.

Hyperbolic Equations

Not all conic sections are ellipses or circles; some are hyperbolas. A hyperbolic equation often presents itself with a characteristic form, such as \(X^2 - Y^2 = K\), where \(K\) is a constant.

• In hyperbolic equations, one variable is subtracted from another rather than both being added, which is typical for ellipses or circles.
• This structure signals the presence of asymptotes, creating the "open ends" of a hyperbola.
• When rotating to new coordinates, if an equation retains this structure (as in the exercise solved), it still represents a hyperbola, now simply oriented differently in the plane.

Understanding hyperbolic equations helps distinguish between various conic sections and explore their unique properties, such as branches opening either horizontally or vertically, depending on the equation's configuration. Conic sections, through transformations, reveal their intrinsic geometric beauty and complexity.