Question

In each of the following cases, determine the counterclockwise rotation of the \(x y\) -axes that brings the conic section into the standard form and determine the conic section. (a) \(11 x^{2}+3 y^{2}+6 x y-12=0\), (b) \(5 x^{2}-3 y^{2}+6 x y+6=0\), (c) \(2 x^{2}-y^{2}-4 x y-3=0\), (d) \(6 x^{2}+3 y^{2}-4 x y-7=0\), (e) \(2 x^{2}+5 y^{2}-4 x y-36=0\).

Short answer

This exercise involves transforming the given equations into standard form and then identifying the conic sections. This involves first calculating the angle of rotation using the formula \(tan(2θ) = B / (A - C)\), then applying a rotation of the axes, and finally identifying the conic based on the equation's resemblance to the standard forms of four conic sections: circle, ellipse, parabola, and hyperbola.

Step-by-step solution

Clarify the Given Equation

Firstly, each given equation is representing a conic section. If it is in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(B ≠ 0\), it means that the axes of the conic section are rotated counterclockwise by an angle θ. This angle θ can be found using \(tan(2θ) = B / (A - C)\). After finding angle θ, to bring it into standard form, rotation of axes is required.

Determine the Rotation Angle

We can find the angle of rotation θ by using the advantage of \(tan(2θ) = B / (A - C)\). In each of the given conic equations, A, B, and C can be identified as the coefficients of \(x^2\), \(xy\) and \(y^2\) respectively. By substituting A, B and C into formula, compute the value of \(2θ\) and thus θ can be calculated. Therefore, we get the counterclockwise rotation of the \(x y\) -axes.

Convert into Standard Form

Substitute the value of angle θ obtained and apply the rotation of axes to convert the conic section into standard form. To rotate the axes, use the transformation equations \(x = x'cosθ - y'sinθ\) and \(y = y'cosθ + x'sinθ\). Substitute these in the original equation. Here, we will have terms containing \(x'y'\). Combine these terms properly and complete the squares to convert the conic into its standard form. Consider first degree terms and complete the squares.

Identify the Conic Section

Once we have transformed the conic equation into standard form, we can identify the type of conic section by comparing it with the standard equation of a parabola, ellipse, hyperbola and circle. For example, if both \(x^2\) and \(y^2\) have the same coefficient, it is a circle; if they have different coefficients but the signs are same, it is an ellipse; if the signs are different, it is a hyperbola; and if only one of \(x^2\) or \(y^2\) exists, it is a parabola.

Key concepts

Standard Form of Conic Sections

The standard form of a conic section is a simplified equation that allows us to easily classify and analyze the curve. Conic sections are the curves obtained by intersecting a plane with a double-napped cone, resulting in circles, ellipses, parabolas, and hyperbolas.

For each of these conics, the standard forms are:

• Circle: \(x-h)^2 + (y-k)^2 = r^2\)
• Ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
• Parabola: \(y-k) = 4p(x-h)^2\) or \(x-h) = 4p(y-k)^2\)
• Hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

Where \(h, k\) are the coordinates of the center, \(r\) is the radius for a circle, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes for an ellipse or hyperbola, and \(p\) is the distance from the vertex to the focus of a parabola.

Transforming a rotated conic section into one of these standard forms involves repositioning the axes or completing the square to eliminate cross-product terms. This simplification is crucial as it reveals the properties of the conic and facilitates easier analysis and graphing.

Rotation of Axes

When a conic section is not aligned with the coordinate axes, we can reorient our perspective through a rotation of axes. This process involves changing the orientation of the coordinate system so that the conic section can be represented without the \(xy\) term that indicates rotation.

To accomplish this, we use the transformation:\begin{align*}x' &= x\cos(\theta) - y\sin(\theta) \y' &= x\sin(\theta) + y\cos(\theta)\end{align*}Where \(x', y'\) are the new coordinates after rotation by angle \(\theta\), and \(x, y\) are the original coordinates. The angle \(\theta\) of rotation is calculated such that it eliminates the \(xy\) term when substituting the new coordinates back into the equation. This results in an equation with only \(x'^2\) and \(y'^2\) terms, possibly accompanied by linear and constant terms.

By applying this rotation, the equation of the conic section can be expressed in the standard form, which is aligned with the new axes. The precise angle for the rotation is found using trigonometry, specifically the tangent of twice the angle, \(\tan(2\theta) = \frac{B}{A - C}\), where \(A, B,\) and \(C\) are coefficients from the original equation.

Identifying Conic Sections

Once we have transformed a conic section into its standard form, identifying the type of conic becomes relatively straightforward.

The general strategy consists of comparing the equation to the standard forms of the four basic conics: circles, ellipses, parabolas, and hyperbolas. The key lies in observing the coefficients and the signs of the squared terms:

• A circle will have equal coefficients for \(x'^2\) and \(y'^2\) with the same sign.
• An ellipse, like a circle, has the same signs for the squared terms, but the coefficients will differ.
• A parabola will feature only one squared term, either \(x'^2\) or \(y'^2\).
• A hyperbola will have squared terms with opposite signs, indicating a subtraction in its equation.

If the equation does not conveniently resemble one of the standard forms, further algebraic manipulation, such as completing the square, may be needed. Additionally, recognizing the presence of certain terms can be indicative of the conic's orientation or location, such as the h and k in the standard forms, which determine the center of the circle, ellipse, or hyperbola. By mastering these identification techniques, students can more readily analyze and graph conic sections, no matter their initial orientation.