Question

Find the largest and smallest distances from the origin to the conic whose equation is \(5 x^{2}-6 x y+5 y^{2}-32=0\) and hence determine the lengths of the semiaxes of this conic.

Short answer

The lengths of the semi-major and semi-minor axes are found by comparing coefficients in the transformed standard form of the ellipse equation.

Step-by-step solution

Identify the Conic

The given equation is \[5 x^{2} - 6 x y + 5 y^{2} - 32 = 0\]. This is a general second-degree equation in the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \].

Determine the Type of Conic

Compute the discriminant, \( \text{D} = B^2 - 4AC \). Here \(A=5\), \(B=-6\), and \(C=5\). Substituting these values: \[ \text{D} = (-6)^2 - 4(5)(5) = 36 - 100 = -64 \]. Since \( \text{D}<0 \), the conic is an ellipse.

Transform the Equation

To find the axes of the ellipse, we need to eliminate the \(xy\)-term. Rotate the coordinate system by an angle \( \theta \) where \[ \tan(2\theta) = \frac{B}{A-C} \rightarrow \tan(2\theta) = \frac{-6}{0} = -\text{undefined} \]. Hence, \( 2\theta = -\frac{\pi}{2} \), or \( \theta = -\frac{\theta}{4} \).

Apply the Rotation

Using the rotation matrix, where \(x = X\frac{\text{sqrt}(2)}{2} - Y\frac{\text{sqrt}(2)}{2}\) and \(y = X\frac{\text{sqrt}(2)}{2} + Y\frac{\text{sqrt}(2)}{2}\), we can express the original equation in new coordinates (\(X, Y\)).

Simplify the Transformed Equation

Substituting these into the equation and simplifying results in an equation of the form \[A'X^2 + C'Y^2 = 32\], after appropriate scaling and simplification, showing the standard form of an ellipse.

Identify the Semi-Axes Lengths

The ellipse equation resembles the form \( \frac{X^2}{a^2} + \frac{Y^2}{b^2} = 1 \). Comparing coefficients gives the lengths of the semi-major and semi-minor axes: \( a = \text{sqrt}(A') \), \( b = \text{sqrt}(C') \).

Determine the Extreme Distances

Maximum distance from the origin to the ellipse corresponds to the length of the semi-major axis, and minimum distance corresponds to the length of the semi-minor axis.

Key concepts

Understanding Conic Sections in Coordinate Geometry

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These curves can take different shapes depending on the angle and position of the intersecting plane. The primary types of conic sections are:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each type has unique properties and equations. In this exercise, we are dealing with an ellipse, which is characterized by its distinctive oval shape. To identify the type of conic, we use the discriminant of the general second-degree equation given by: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). You can determine the conic section using the discriminant \(D = B^2 - 4AC\). For this problem, the discriminant is negative \(D < 0\), indicating an ellipse.

Ellipse Equation and Its Standard Form

The equation of an ellipse in its standard form is written as: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Here, \( (h, k) \) represents the center of the ellipse, while \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The semi-major axis is the longest radius of the ellipse, and the semi-minor axis is the shortest.
In the given exercise, we start with the equation \(5 x^{2} - 6 x y + 5 y^{2} - 32 = 0\). To find its standard form, eliminate the mixed \(xy\)-term by rotating the coordinate system. Once transformed, the equation simplifies to \(A'X^2 + C'Y^2 = 32\), revealing the standard ellipse form: \(\frac{X^2}{a^2} + \frac{Y^2}{b^2} = 1\). Identifying these coefficients helps pinpoint the distances from the origin.

Determining Semi-Major and Semi-Minor Axes

The lengths of the semi-major and semi-minor axes of an ellipse are crucial to understanding its geometric properties. These lengths are derived from the coefficients in the standard form equation. For an ellipse given by \(\frac{X^2}{a^2} + \frac{Y^2}{b^2} = 1\), \(a\) and \(b\) represent the semi-major and semi-minor axes:

• \(a\) is the larger value and corresponds to the semi-major axis.
• \(b\) is the smaller value and corresponds to the semi-minor axis.

In the context of the given problem, after transforming and simplifying, we find the form \(A'X^2 + C'Y^2 = 32\). Therefore, the lengths of the axes can be determined by comparing the coefficients: \(a = \sqrt{A'}\) and \(b = \sqrt{C'}\). These lengths help pinpoint the maximum and minimum distances from the origin to the ellipse.

Rotation of Axes to Simplify Ellipse Equations

The presence of an \(xy\)-term in the ellipse equation often necessitates the rotation of coordinate axes to simplify. This eliminates the mixed term, helping convert the equation to its standard form. The rotation angle \(\theta\) is determined using: \(\tan(2\theta) = \frac{B}{A - C}\).
In the discussed exercise, substituting \(B = -6\), \(A = 5\), and \(C = 5\) yields \(\tan(2\theta) = \frac{-6}{0} = -\text{undefined}\), implying \(2\theta = -\frac{\pi}{2}\) or \(\theta = -\frac{\pi}{4}\).
Using rotation matrices \(x = X\frac{\sqrt{2}}{2} - Y\frac{\sqrt{2}}{2}\) and \(y = X\frac{\sqrt{2}}{2} + Y\frac{\sqrt{2}}{2}\), we can rewrite the original equation in new coordinates (\(X, Y\)). This transformation simplifies the equation to a recognizable ellipse equation, which can be further analyzed to determine the semi-major and semi-minor axes.