Problem 50 - Advanced Engineering Mathematics

Question

From \(x=2 \cos 2 t-2 \sin 2 t, y=-\cos 2 t\) we find \(x+2 y=-2 \sin 2 t .\) Then \((x+2 y)^{2}=4 \sin ^{2} 2 t=4\left(1-\cos ^{2} 2 t\right)=4-4 \cos ^{2} 2 t=4-4 y^{2}\) and \(x^{2}+4 x y+4 y^{2}=4-4 y^{2} \quad\) or \(\quad x^{2}+4 x y+8 y^{2}=4\) This is a rotated conic section and, from the discriminant \(b^{2}-4 a c=16-32 < 0,\) we see that the curve is an ellipse.

Step-by-Step Solution

Answer

The curve is an ellipse.

Step 1: Given Parametric Equations

We start with the parametric equations given by: \( x = 2\cos 2t - 2\sin 2t \) and \( y = -\cos 2t \). We need to manipulate these equations to find relationships and properties of the geometric figure they describe.

Step 2: Calculate \( x + 2y \)

Substitute the expressions for \(x\) and \(y\) into \(x + 2y\) to get \(x + 2y = 2\cos 2t - 2\sin 2t + 2(-\cos 2t)\). This simplifies to \(x + 2y = -2\sin 2t\).

Step 3: Square the Expression

Square the expression \(x + 2y = -2\sin 2t\) to get \((x + 2y)^2 = 4\sin^2 2t\). Use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta \) to re-write this as \(4 (1 - \cos^2 2t) = 4 - 4\cos^2 2t\).

Step 4: Express in terms of \(y\)

Recall that \(y = -\cos 2t\) so \(\cos^2 2t = y^2\). Substitute to obtain \(4 - 4\cos^2 2t = 4 - 4y^2\).

Step 5: Equation of Conic Section

We have the identity \( x^2 + 4xy + 8y^2 = 4 \) which is derived by observing that \( x^2 + 4xy + 4y^2 = 4 - 4y^2 \cdot 4y^2 + 4y^2 \). Add \(4y^2\) to both sides to achieve the final result.

Step 6: Determine Type of Conic

The expression \( x^2 + 4xy + 8y^2 = 4 \) is a conic section. To determine its type, calculate the discriminant \(b^2 - 4ac = 16 - 32 = -16\). Since this is less than zero, the conic section is an ellipse.

Key Concepts

Parametric Equations

Parametric equations are an essential tool in mathematics, especially when dealing with curves and motion. Typically, these equations express the coordinates of the points that make up a geometric figure as functions of one or more variables known as parameters. They are incredibly useful when describing complex curves because they provide a way to represent both the path and the position on that path at any given time or parameter value.

In the original exercise, we encountered the parametric equations:

• \( x = 2\cos 2t - 2\sin 2t \)
• \( y = - \cos 2t \)

Here, the parameter is \(t\), which is often used to represent time or a similar variable. By manipulating and converting these parametric equations, one can explore different representations of the same curve, like turning them into a single Cartesian equation. This enables us to analyze the shape, size, and position of the curve in a standard coordinate system.

Ellipses

Ellipses are a type of conic section that arise frequently in geometry. They can be imagined as stretched circles, defined by the set of points such that the sum of the distances to two fixed points (known as foci) is constant. This shape is common in a variety of real-world applications including planetary orbits and architectural designs.

In the exercise, after determining the equation from the given parametric equations, we calculated:

• \( x^2 + 4xy + 8y^2 = 4 \)

which is the standard form of a conic section. To confirm that it is an ellipse, we looked at essential characteristics like symmetry and the discriminant calculation. The discriminant revealed that the resulting figure is an ellipse, but more on that in the next section.

Understanding how parametric equations can describe an ellipse is fundamental when transitioning from theoretical mathematics to practical applications, such as calculating orbital paths or designing objects in engineering. This knowledge bridges the gap between abstract math and concrete scenarios.

Discriminant

The discriminant is a critical number used in algebra to determine the nature and type of conic sections, including ellipses, parabolas, and hyperbolas. It is derived from the general quadratic equation for conics, which has the form: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

The discriminant for conic sections is calculated using the formula \(b^2 - 4ac\). Depending on its value:

• If \(b^2 - 4ac > 0\), the conic is a hyperbola.
• If \(b^2 - 4ac = 0\), the conic is a parabola.
• If \(b^2 - 4ac < 0\), the conic is an ellipse, as in our exercise.

In our case, from \(x^2 + 4xy + 8y^2 = 4\), we computed the discriminant \(16 - 32 = -16\). Since this value is less than zero, it conclusively indicates that the conic is an ellipse.

The discriminant simplifies the process of identifying the type of conic without needing to graph it or perform complex transformations. This is particularly valuable when dealing with complex or non-standard orientations of conic sections.