Problem 2 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}-18 x y+27 y^{2}-5 x+7 y=-4\)

Step-by-Step Solution

Answer

The discriminant is 0, indicating a degenerate conic section.

Step 1: Identify the coefficients

The equation given is \(3x^{2}-18xy+27y^{2}-5x+7y=-4\). From this equation, we can identify the coefficients as follows: \(A = 3\), \(B = -18\), and \(C = 27\). These coefficients are associated with the terms \(x^2\), \(xy\), and \(y^2\) respectively.

Step 2: Calculate the Discriminant

The discriminant for the conic section is calculated using the formula \(B^2 - 4AC\). Substituting the values of \(A\), \(B\), and \(C\) into the formula, we get: \(B^2 - 4AC = (-18)^2 - 4 \cdot 3 \cdot 27\).

Step 3: Compute Discriminant Value

Calculate \((-18)^2\) which is \(324\), and \(4 \cdot 3 \cdot 27\) which is \(324\). Substituting these values back, we have: \(324 - 324 = 0\).

Step 4: Classify the Conic Section

According to the discriminant analysis, if \(B^2 - 4AC = 0\), the equation represents a degenerate conic section. For non-degenerate cases, this specific form with \(B^2 - 4AC = 0\) typically represents a parabola in non-degenerate form, but here it suggests degeneracy due to equal \(A\), \(B\), and \(C\).

Key Concepts

Discriminant

The discriminant is a crucial element when studying conic sections, as it helps to determine the type of conic section an equation represents. Given a quadratic equation in two variables, the discriminant is calculated using the formula \( B^2 - 4AC \). This formula comes in handy when you have coefficients associated with terms like \(x^2\), \(xy\), and \(y^2\). Here,

• \(A\) is the coefficient of \(x^2\),
• \(B\) is the coefficient of \(xy\),
• and \(C\) is the coefficient of \(y^2\).

The value of the discriminant tells you which type of conic section you're dealing with:

• If \(B^2 - 4AC < 0\), it's generally an ellipse.
• If \(B^2 - 4AC = 0\), it's typically a parabola.
• If \(B^2 - 4AC > 0\), it's a hyperbola.

In degenerate cases, like when the value is zero and other conditions hold, special forms may emerge, which might not classify perfectly into one of these three categories.

Parabolas

Parabolas are unique conic sections that can be identified through their discriminant value equaling zero \((B^2 - 4AC = 0)\). They have distinctive properties that set them apart:

• They have a single curve that extends infinitely in both directions.
• Often described as the path of a moving point equidistant from a fixed point, called the focus, and a line, called the directrix.
• They find practical use in areas like satellite dishes and flashlight reflectors due to their reflective properties.

In some cases, a discriminant of zero might suggest degeneration, meaning it could potentially not represent an actual arc or one of the standard conic sections.

Ellipses

Ellipses are elegant conic sections that can be identified when the discriminant is less than zero \((B^2 - 4AC < 0)\). An ellipse appears as an elongated circle, and it has several interesting properties:

• Foci: Every ellipse has two fixed points on its interior.
• Symmetry: It is symmetric about its major and minor axes.
• Circular Potential: Both circles and ellipses belong to this family, with a circle being a special case where the two axes are equal.

Ellipses are not just mathematical curiosities; they play a key role in the orbits of planets and satellites, which follow elliptical paths.

Hyperbolas

Hyperbolas are other fascinating shapes that fall under conic sections. When the discriminant is greater than zero \((B^2 - 4AC > 0)\), you're looking at a hyperbola. Here's why they stand out:

• Two Curves: They consist of two disconnected curves, known as branches.
• Asymptotes: They have asymptotes—lines which the hyperbola approaches but never touches or intersects.
• Rectangular Constructions: In mathematics and physics, hyperbolas are often used to describe situations involving exponential growth and decay.

Hyperbolas might seem tricky at first, but they hold significant importance in various fields, from engineering to astronomy.