Question

The equation of a particular conic section is $$ Q \equiv 8 x_{1}^{2}+8 x_{2}^{2}-6 x_{1} x_{2}=110 $$ Determine the type of conic section this represents, the orientation of its principal axes, and relevant lengths in the directions of these axes.

Short answer

The conic section is an ellipse. Its principal axes have lengths approximating 1 unit and 1.5 units.

Step-by-step solution

Recognize the General Form

The given equation is \[ Q \textbackslash equiv 8 x_{1}^{2}+8 x_{2}^{2}-6 x_{1} x_{2}=110 \] Recognize that this equation is in the general quadratic form for conic sections: \[ Ax_{1}^{2} + Bx_{1}x_{2} + Cx_{2}^{2} = D \]

Identify Coefficients

In the given equation, identify the coefficients:\[ A = 8, \quad B = -6, \quad C = 8, \quad D = 110 \]

Use the Discriminant Test

Use the discriminant of the quadratic form to determine the type of conic section. The discriminant \( \textbackslash Delta \) is given by \[ B^2 - 4AC \]. Substitute in the coefficients:\[ \textbackslash Delta= (-6)^{2} - 4(8)(8) = 36 - 256 = -220 \]Since the discriminant is negative (\( \textbackslash Delta < 0 \)), the conic section is an ellipse.

Diagonalize the Quadratic Form

Diagonalize the quadratic form by finding its eigenvalues and eigenvectors. Write the quadratic portion as a matrix \[ \begin{pmatrix} 8 & -3 \ -3 & 8 \end{pmatrix} \].Solve the eigenvalues through \[ \textbackslash| A - \textbackslash lambda I \textbackslash| = 0 \], where \[ \textbackslash| 8 - \textbackslash lambda & -3 \ -3 & 8 - \textbackslash lambda \textbackslash| = 0 \].Finding the determinant gives us the characteristic equation:\[ (8 - \textbackslash lambda)^2 - 9 = 0 \].Solving for \( \textbackslash lambda \) gives:\[ \textbackslash lambda_1 = 11, \quad \textbackslash lambda_2 = 5 \].The eigenvectors can be used to find the major and minor axes.

Find Relevant Lengths

The lengths along the principal axes can be determined by relating to the eigenvalues. They correspond to the intersections of the ellipse with the principal axes:\[ \frac{110}{\textbackslash lambda_1} \quad \textbackslash text{and} \quad \frac{110}{\textbackslash lambda_2} \].Thus, the lengths are \[ \sqrt{\frac{110}{11}} \quad \textbackslash text{and} \quad \sqrt{\frac{110}{5}} \].Calculating these lengths gives:\[ \frac{\textbackslash sqrt{10}}{\textbackslash sqrt{11}} \approx \frac{\textbackslash sqrt{30}}{\textbackslash sqrt{5}} \].Simplified, the lengths along the principal axes are approximately 1 unit and \textbackslash sqrt2.985

Key concepts

Ellipse

An ellipse is a shape that looks like a stretched circle. Imagine a circle that's been pulled from two sides. In mathematics, it appears often as a conic section - shapes formed by slicing through a cone. Ellipses have two main components: the major axis (the longer one) and the minor axis (the shorter one). The lengths of these axes are important for many calculations. When you look at an ellipse equation, such as the one given in the problem, its general form is \(Ax_1^2 + Bx_1x_2 + Cx_2^2 = D\). Understanding the nature of its coefficients can tell you a lot about the ellipse itself. Additionally, ellipses have some unique properties such as uniform curvature and being centered around two focus points. These properties are often used in fields ranging from astronomy to engineering.

Quadratic Form

Quadratic forms are expressions involving multiple variables squared. For conic sections, they play a crucial role. The general quadratic form is \(Ax_1^2 + Bx_1x_2 + Cx_2^2 = D\). Here, the coefficients (A, B, C) provide information about the curve's orientation and type. For example, if B = 0, the axes of symmetry align with the coordinate axes. However, if B ≠ 0, the curve might be rotated. This quadratic representation helps in determining whether the conic section is a circle, ellipse, parabola, or hyperbola. To simplify such forms, we often diagonalize the expression by finding eigenvalues and eigenvectors, making it easier to understand the nature and orientation of the shape.

Eigenvalues

Eigenvalues are special numbers associated with matrices, which in turn describe our quadratic forms. For the given problem, the matrix representing the quadratic form is \(\begin{pmatrix} 8 & -3 \ -3 & 8 \end{pmatrix}\). To find the eigenvalues, we solve the characteristic equation \(\|A - \lambda I\| = 0\). This involves calculating the determinant of [8 - λ & -3 ; -3 & 8- λ], setting it to zero and solving for λ. The roots of this equation, λ1 and λ2, are our eigenvalues. They provide essential insights into the conic section’s properties. Specifically, they help us determine the lengths and orientation of the principal axes of the ellipse. Higher eigenvalues usually correspond to the major axis while lower ones to the minor axis.

Discriminant

The discriminant of a quadratic form helps identify the type of conic section. It’s calculated using \(B^2 - 4AC\). For the given problem, we substitute the coefficients A = 8, B = -6, and C = 8 to get ∆ = (-6)² - 4(8)(8), which equals -220. The negative value indicates that the conic section is an ellipse. This test is easy and effective. If the discriminant had been positive, the section would be a hyperbola. If it were zero, it would be a parabola. This quick check helps confirm the nature of the conic section we are dealing with.

Principal Axes

Principal axes are the main directional lines of an ellipse. They intersect at the center of the ellipse and are perpendicular to each other. The longest one is called the major axis, and the shorter one is the minor axis. Determining these axes involves diagonalizing the quadratic form matrix and finding its eigenvectors. The directions of these eigenvectors indicate the orientation of the principal axes. For our problem, solving for the eigenvalues, we get λ1 = 11 and λ2 = 5. Using these values, we can determine the axes’ lengths by calculating \(\sqrt{\frac{110}{\lambda_1}}\) and \(\sqrt{\frac{110}{\lambda_2}}\). Hence, the lengths related to the principal axes provide crucial information about the ellipse's scale and dimensions.