Problem 311 - Calculus Volume 3

Question

Determine which of the conic sections is described. $$ 52 x^{2}-72 x y+73 y^{2}+40 x+30 y-75=0 $$

Step-by-Step Solution

Answer

The conic section is an ellipse.

Step 1: Identify the General Form

The general form for a conic section is given by \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A=52\), \(B=-72\), and \(C=73\). We need to find the type of conic based on this equation.

Step 2: Calculate the Discriminant

The discriminant \(D\) for conic sections is calculated as \(D = B^2 - 4AC\). Plugging in the values, we have \(D = (-72)^2 - 4(52)(73)\).

Step 3: Simplify the Discriminant

Calculate \((-72)^2 = 5184\) and \(4 \times 52 \times 73 = 15184\). Thus, \(D = 5184 - 15184 = -10000\).

Step 4: Determine the Type of Conic Section

For a conic section, if \(D < 0\), the equation represents an ellipse. Therefore, since \(D = -10000 < 0\), this conic is an ellipse.

Key Concepts

Ellipse

An ellipse is a plane curve that forms a symmetrical, oval shape. It looks similar to a stretched circle. Ellipses have two focal points, often simply known as "foci." The key characteristic of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant.
Ellipses have two axes - the major axis and the minor axis. The major axis is the longer one, while the minor axis is shorter. The center of the ellipse is located where these two axes intersect.
In terms of algebra, if we determine that the discriminant from the general form equation of a conic section is less than zero, we identify the conic as an ellipse. This is because the discriminant, which involves calculating values based on the coefficients of the equation, suggests an ellipsoidal shape predominantly when negative.

Discriminant

The discriminant of a conic section's general equation helps to determine what type of conic section it represents. This is calculated using the formula \(D = B^2 - 4AC\).The variables \(A\), \(B\), and \(C\) are coefficients from the quadratic terms of the general form equation.
The discriminant indicates:

• If \(D > 0\), the conic section is a hyperbola.
• If \(D = 0\), it represents a parabola.
• If \(D < 0\), the conic is an ellipse.

In our case, the discriminant is \(-10000\). This negative number confirms the classified conic shape as an ellipse. Therefore, the discriminant is a vital tool in geometry for quickly identifying conic sections without graphing them.

General Form Equation

The general form equation for conic sections is expressed as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\),where each letter represents a coefficient associated with the equation's terms.
This general form can depict any conic section: ellipse, parabola, or hyperbola, depending on the relationship between \(A\), \(B\), and \(C\).

• In an ellipse, the discriminant \(B^2 - 4AC\) is less than zero.
• The coefficient \(B\) is crucial when determining the conic's orientation and angle.

Understanding this general form allows us to analyze and identify conic sections algebraically. By manipulating this form, we can classify the conic type, as the exercise requires, based on the discriminant.