Problem 23 - Precalculus with Limits

Question

In Exercises 13-26, rotate the axes to eliminate the \(xy\)-term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. \(3x^2 -2\sqrt{3}xy+y^2+2x+2\sqrt{3}y = 0\)

Step-by-Step Solution

Answer

First, compute for the rotation angle, then rotate the axes through substitution. Simplify the new equation, after which the graph of the resulting equation is plotted. This process eliminates the \(xy\)-term in the equation and results in an equation in standard form.

Step 1: Compute the Rotation Angle

Compute the rotation angle using the formula \(\theta = \frac{1}{2} \cdot arctan \left(\frac{-b}{a-c} \right)\), where \(b\) is the coefficient of \(xy\), \(a\) is the coefficient of \(x^2\), and \(c\) is the coefficient of \(y^2\). So, from the equation \(3x^2 -2\sqrt{3}xy+y^2+2x+2\sqrt{3}y = 0\), \(a = 3\), \(b = -2\sqrt{3}\), and \(c = 1\). Substituting these values into the equation, we get \(\theta = \frac{1}{2} \cdot arctan \left(\frac{-(-2\sqrt{3})}{3-1} \right) = \frac{1}{2} \cdot arctan (\sqrt{3}) = \frac{\pi}{6}\).

Step 2: Rotate the Axes

The rotation of axes is through substitution. We can rotate the axes using the formula \(x = rcos(\theta - \phi)\) and \(y = rsin(\theta - \phi)\), where \(r\) and \(\phi\) are the polar coordinates of \(x\) and \(y\). Let \(x' = xcos(\pi/6) + ysin(\pi/6)\) and \(y' = -xsin(\pi/6) + ycos(\pi/6)\), which simplifies to \(x' = (x + y)/\sqrt{2}\) and \(y' = (-x + y)/\sqrt{2}\). So, \(x = (x' + y')/\sqrt{2}\) and \(y = (-x' + y')/\sqrt{2}\). Replace these in the original equation, and simplify.

Step 3: Simplify the new equation

After substituting the values back into the equation and simplifying, the new equation in terms of \(x'\) and \(y'\) should have no mixed term (i.e. \(x'y'\)). The new equation should be in the standard form: \(Ax'^2 + By'^2 + Cx' + Dy' = E\), which describes a conic section graph.

Step 4: Sketch the Graph

Graph the resulting equation in the \(x'\)-\(y'\) plane. Also, show the old axes in the graph to visualize the rotation of the axes.

Key Concepts

Conic Sections

When exploring the world of geometry, conic sections are intriguing shapes that emerge from slicing a cone with a plane. Envision a right circular cone, with another identical cone attached to it base-to-base. Different intersections of this structure with a plane result in different conic sections: parabolas, ellipses, circles, and hyperbolas.

Depending on the angle and the distance of the intersecting plane to the cone's tip, these unique shapes come into existence. In algebraic terms, conic sections can be represented by quadratic equations of the form Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, with the coefficients 'A', 'B', and 'C' dictating the type of the section.

Polar Coordinates

In a planar coordinate system, as a complement to the familiar Cartesian (x, y) coordinates, we have polar coordinates that define a point's position based on its angle and distance from a reference point, often called the pole, akin to the North Pole on our Earth. Here, the angle (usually represented as \( \theta \)) measures how far around from the polar axis (an equivalent to longitude in global terms) a point is, and the radius (designated as \( r \) ) signifies how far from the pole the point lies.

In mathematics, polar coordinates (\( r, \theta \) ) offer an alternative method to describe the location of points in a plane, particularly useful in scenarios involving circular or spherical symmetry, such as in the equations of conic sections.

Rotation of Axes

The concept of rotating axes involves visualizing the shift of the entire coordinate system around a single point, without physically moving the figure itself. It's akin to turning the grid under a stationary object rather than spinning the object. This technique is often employed to simplify the equations of conic sections.

By rotating the axes by a certain angle — calculated to eliminate the mixed \( xy \) term in the equation of a conic — we typically transition from a more complex equation to one of standard form. Picture this as untwisting a tight rope; the tension represents the complexity added by the \( xy \) term, and rotating the axes releases this tension, straightening the equation into a more manageable shape.

Standard Form of Equation

The elegance of mathematics becomes particularly evident when equations are expressed in their standard form, which is the simplest and most informative version. For conic sections, the standard forms (after eliminating the \( xy \) term by rotation of axes) align to structures like \( Ax^2 + By^2 + Cx + Dy = E \) for each specific conic. This sleek equation permits us to discern the nature of the conic merely by inspecting the coefficients.

For instance, if 'A' and 'B' have the same sign and are not zero, the equation could represent an ellipse or a circle, depending on whether 'A' equals 'B'. Conversely, if 'A' and 'B' have opposite signs, we may be gazing upon a hyperbola. The process of transforming an equation into its standard form not only furnishes us with the ability to sketch the graph effortlessly but also unveils the characteristics and behavior of the conic section.