Problem 29 - Precalculus: Mathematics for Calculus

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$2 x^{2}-4 x y+2 y^{2}-5 x-5=0$$

Step-by-Step Solution

Answer

The conic is a parabola.

Step 1: Identify Parameters in General Conic Form

The given equation is \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). For conics in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), compare to find: \(A = 2\), \(B = -4\), and \(C = 2\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) for conics is given by \(\Delta = B^2 - 4AC\). Substitute the values: \((-4)^2 - 4(2)(2) = 16 - 16 = 0\).

Step 3: Interpret the Discriminant

The discriminant \(\Delta = 0\). This indicates that the conic is a parabola.

Step 4: Graph the Conic

Using a graphing device, plot the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). Observe the shape and confirm it visually represents a conic section consistent with the classification as a parabola.

Key Concepts

Understanding Parabolas

A parabola is a type of conic section that can be easily recognized as a U-shaped curve on the graph.
It is defined mathematically as a set of points that are equidistant from a fixed point called the "focus" and a line known as the "directrix."
Parabolas are commonly found in quadratic equations, represented by the formula:

• The general quadratic form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
• For a parabola, the discriminant \[ \Delta = B^2 - 4AC = 0 \]

In our problem, the discriminant calculated was zero, which confirms the conic section as a parabola.
Parabolas are fascinating because they appear in nature and technology, like the path of a thrown ball or in satellite dishes, due to their reflective properties.

Exploring Conic Sections

Conic sections are curves obtained by intersecting a plane with a double napped cone.
The resulting shapes depend on the angle and position of the intersection. They are pivotal in both algebra and geometry.

• The four primary types of conic sections are:
  • Circle
  • Ellipse
  • Parabola
  • Hyperbola
• The type of conic section is determined using the discriminant formula: \[ \Delta = B^2 - 4AC \]
• The values of \[ \Delta \] determine the conic:
  • If \[ \Delta > 0 \], the conic is a hyperbola.
  • If \[ \Delta = 0 \], it's a parabola.
  • If \[ \Delta < 0 \] and \[ A = C \], it's a circle.
  • If \[ \Delta < 0 \] and \[ A eq C \], it's an ellipse.

Using these rules, we can determine that the given equation represents a parabola.

Graphing Conics

Graphing conic sections helps visually confirm their type based on their geometric properties.
Using tools like graphing calculators or software, we can plot the equation and analyze the outcome.

• For a parabola:
  • If the graph appears as a U-shape, it confirms the conic is a parabola.
  • To graph, rearrange the equation into a form that identifies its vertex and direction.
• Steps to graph other conic sections:
  • Circles appear as perfect loops.
  • Ellipses form elongated ovals.
  • Hyperbolas appear as two opposing curved branches.

For the given problem, plotting on a graph verifies our classification, highlighting its parabolic nature.