Problem 14 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$-3 x^{2}+3 \sqrt{3} x y-4 y^{2}+9=0$$

Step-by-Step Solution

Answer

The conic section represented by the equation is an ellipse.

Step 1: Identify the General Form

The general form of a conic section equation is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here, we have the equation \( -3x^2 + 3\sqrt{3}xy - 4y^2 + 9 = 0 \). Comparing with the general form, we identify \( A = -3 \), \( B = 3\sqrt{3} \), and \( C = -4 \).

Step 2: Compute the Discriminant

The discriminant \( \, \Delta = B^2 - 4AC \, \) helps determine the type of conic section. Substituting the values, we calculate \( \, \Delta = (3\sqrt{3})^2 - 4(-3)(-4) = 27 - 48 = -21 \, \).

Step 3: Determine the Type of Conic Section

The value of the discriminant \( \Delta = -21 \, \) indicates the type of conic section: if \( \Delta > 0 \), it's a hyperbola; if \( \Delta = 0 \), it's a parabola; if \( \Delta < 0 \), it's an ellipse. Here, since \( \Delta < 0 \), the conic section is an ellipse.

Key Concepts

Ellipse

Let's dive into the concept of an ellipse, a fascinating shape in Conic Sections. An ellipse looks like an elongated circle, kind of like the path planets take around the sun.
An ellipse has several interesting properties, but at its core, it is defined as the set of all points such that the sum of the distances to two fixed points (called foci) is constant.

• The longer axis is called the major axis, and the shorter one is the minor axis.
• The center is the midpoint between the two foci.

If you picture a racetrack, the shape of that track resembles an ellipse. Understanding ellipses is vital because they have real-world applications like planetary orbits and optical systems. In equations, ellipses often appear with terms involving both squared terms of the variables involved, leading to equations of the form \( Ax^2 + Cy^2 + ... = 0 \), which differentiates them from circles where \( A = C \).

Discriminant in Conic Sections

The discriminant is a powerful tool in mathematics, especially for identifying conic sections from their general equations. It is derived from the coefficients of the terms in the conic's equation. The discriminant is calculated using the formula \( B^2 - 4AC \).

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse.

For example, in our original exercise, after calculating the discriminant as \( -21 \), a negative value, we confirm that the conic section is an ellipse. Understanding how to compute and interpret the discriminant is crucial because it allows us to quickly classify conic sections and understand their geometric properties.

General Form of Conic Sections

The general form of a conic section's equation is essential for determining the type of conic we are dealing with. It is generally expressed as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). Here, \( A \), \( B \), and \( C \) are coefficients that determine the conic's type while \( D \), \( E \), and \( F \) can move the conic around the plane.

• When \( B = 0 \) and \( A = C \), the conic is a circle.
• The presence of the \( Bxy \) term can indicate rotation of the conic section.

By identifying these coefficients, especially \( A \), \( B \), and \( C \), we can use the discriminant to determine the specific conic section. Recognizing this form gives us a strategic starting point in analytic geometry to classify the curve and explore its properties further in applications like optics, physics, and engineering.