Question

For each of the following, write the equation in terms of new variables so that it is in standard position, and identify the curve. a. \(x y=1\) b. \(3 x^{2}-4 x y=2\) c. \(6 x^{2}+6 x y-2 y^{2}=5\) d. \(2 x^{2}+4 x y+5 y^{2}=1\)

Short answer

a: Hyperbola; b: Hyperbola; c: Ellipse; d: Ellipse.

Step-by-step solution

Identify the Transformation

For each equation, we must transform the variables so that the quadratic equations are in standard form. This typically involves using a rotation transformation to eliminate the mixed term (xy) where needed.

Rewrite Part (a)

The equation is \(xy=1\). There is no need to transform variables since there is no mixed term and this equation represents a hyperbola.

Transform Equation Part (b)

Equation is \(3x^2 - 4xy = 2\). Use the rotation formulas to eliminate xy: substitution \(x = X \cos{\theta} - Y \sin{\theta}\) and \(y = X \sin{\theta} + Y \cos{\theta}\), where \(\tan{(2\theta)} = \frac{B}{A-C}\).

Determine Angle for Part (b)

For \(3x^2 - 4xy = 2\), A = 3, B = -4, C = 0. \(\tan{(2\theta)} = \frac{-4}{3-0} = -\frac{4}{3}\). Solve for \(\theta\) to rewrite the equation without xy.

Complete Transformation for Part (b)

After calculation, once we choose \(\theta\) to make the xy term zero, rewrite the equation. This step involves computing and solving \(\theta\) to find respective rotated axes; however, look for integratable simplifications.

Identify Curve Type for Part (b)

For part (b), once \(\theta\) is found, the transformed equation depicts the conic section type; here, simplify and identify if its a hyperbola, ellipse, or parabola after rotation.

Repeat Rotation for Part (c)

Equation \(6x² + 6xy - 2y² = 5\). Solve using \(\tan{2\theta} = \frac{6}{6-(-2)} = \frac{6}{8}\), leading to \(\tan{2\theta} = \frac{3}{4}\). Solve \(\theta\), substitute to eliminate the xy term.

Identify Curve Type for Part (c)

After eliminating the xy term, convert and simplify the equation to its simplest form to deduce the type, such as a hyperbola or ellipse.

Rotation for Part (d)

Equation: \(2x² + 4xy + 5y² = 1\). Calculate \(\tan{2\theta} = \frac{4}{2-5} = -\frac{4}{3}\). Solve \(\theta\) appropriately, eliminating the xy term using new X,Y coordinates.

Determine Type for Part (d)

Upon rotation, and simplification eliminate the xy cross-term, identify conic type of transformed equation by finding its canonical form.

Key concepts

Quadratic Equations

Quadratic equations form the basis for analyzing conic sections. These are polynomial equations of degree two and can be expressed in the general form: \[ ax^2 + bxy + cy^2 + dx + ey + f = 0 \]The presence of the product of variables, such as the term \(bxy\), indicates a more complex relation and may require transformations to simplify it. Quadratics can represent different conic sections based on their coefficients and the zero-condition of some variables.

Variable Transformation

Variable transformation is a mathematical technique used to simplify equations or identify their inherent forms by substituting the original variables with new ones. For a quadratic equation with a mixed term like \(xy\), a common method of transformation is the rotation of axes. While often used in solving conic-related problems, transformation can also simplify understanding the geometry through more familiar terms. It involves systematically replacing variables to either remove unwanted terms or emphasize certain characteristics of the equation.

Rotation of Axes

When dealing with conic sections that include a mixed term (\(bxy\)), rotating the axes is crucial. This transformation is mathematically expressed through the formulas:

• \(x = X \cos{\theta} - Y \sin{\theta}\)
• \(y = X \sin{\theta} + Y \cos{\theta}\)

The angle \(\theta\) is determined by solving \(\tan(2\theta) = \frac{B}{A-C}\). By choosing the appropriate \(\theta\), we aim to cancel the \(xy\) term, typically transforming the equation into a more recognizable conic section signature without cross-terms, which simplifies the identification process.

Hyperbola

A hyperbola is one of the primary conic sections characterized by its distinct structure. It can often be recognized after transforming an equation into its standard form, such as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This form is particularly associated with hyperbolas due to one variable being positive and the other negative, indicating two separate branches that mirror each other. The transformational process, especially axis rotation, often reveals this form when the mixed term \(xy\) is eliminated.

Ellipse

Ellipses represent another type of conic section, known for their oval shape. In standard form, their equations appear as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), signifying both squared variables with positive coefficients. Unlike hyperbolas, ellipses do not possess two distinct parts but rather one continuous loop. Identifying an ellipse often involves ensuring that after rotation and elimination of the \(xy\) term, the result maintains two positive squared terms with no negatives involved.

Parabola

Parabolas are unique conic sections with an easily recognizable trait – they contain only one quadratic term associated with either \(x\) or \(y\), never both. A standard form of a parabolic equation can be simplified to look like \(y = ax^2 + bx + c\). Rotation is generally unnecessary for simple parabolas since their structure doesn't involve an \(xy\) term. These curves are often found in practical applications due to their ability to direct paths predictably, such as in projectile motion or reflectors.