Problem 13 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(x^{2}-3 x y+3 y^{2}+6 y=7\)

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the coefficients

The given equation is \(x^2 - 3xy + 3y^2 + 6y = 7\). Re-arrange it to match the general second-degree form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Hence, \(A = 1\), \(B = -3\), \(C = 3\), \(D = 0\), \(E = 6\), \(F = -7\).

Step 2: Calculate the discriminant

The discriminant \(B^2 - 4AC\) is calculated using the coefficients from Step 1: \((-3)^2 - 4(1)(3)\). This simplifies to \(9 - 12 = -3\).

Step 3: Interpret the discriminant

The discriminant \(B^2 - 4AC = -3\) is less than 0, which indicates the equation represents an ellipse.

Key Concepts

Parabolas

A parabola is one of the basic types of conic sections, which are U-shaped graphs that can open either vertically or horizontally. Parabolas can be represented by equations of the form \[ y = ax^2 + bx + c \] for vertical parabolas, or \[ x = a(y - k)^2 + h \] for horizontal ones.
This distinctive shape is thanks to their geometric definition: a set of all points equidistant from a particular point (focus) and a line (directrix).
The vertex, a special point on the parabola, marks the turning point of the curve.

• If you encounter a second-degree equation with no cross-product term (the term involving both x and y, like Bxy), then you're likely dealing with a parabola.
• The axis of symmetry splits the parabola into two mirror-image halves.

An example can be observing the trajectory of a projectile, which typically follows a parabolic path due to gravity.

Ellipses

Ellipses are another type of conic section, characterized by their elongated, oval shape. The general equation for an ellipse in a second-degree form is \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] with condition \[ B^2 - 4AC < 0 \].
This inequality reflects a negative discriminant, which confirms the presence of an ellipse as seen in our example solution.
Ellipses have two focal points (or foci) inside them, and any point on the ellipse has a constant total distance to these foci.

• If the length across the foci (major axis) is longer than the length perpendicular (minor axis), the ellipse appears stretched.
• In special cases, when the foci coincide, the ellipse becomes a circle.

One famous example of an ellipse is the path of planets around the Sun, as per Kepler's first law of planetary motion.

Hyperbolas

A hyperbola is a conic section formed by intersecting a double cone and cutting through both nappes. Its equation generally looks like \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] with the discriminant condition \[ B^2 - 4AC > 0 \].
This positive discriminant reveals two separate branches, unlike the connected shapes of parabola and ellipse.
Hyperbolas have two asymptotes, lines that the branches approach but never touch. These lines help define the hyperbola's shape and orientation.

• Each branch of a hyperbola is mirror-symmetrical over its center point.
• The transverse axis is the line segment joining the vertices.

This concept is frequently applied in fields like engineering, for satellite dishes and radar systems, due to their reflective properties.

Discriminant

The discriminant plays a crucial role in identifying the type of conic section an equation represents. For any second-degree equation of the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \], the discriminant is calculated as \[ B^2 - 4AC \].
Here's how the value determines the conic section:

• If \[ B^2 - 4AC = 0 \], then the equation represents a parabola.
• If \[ B^2 - 4AC < 0 \], the equation indicates an ellipse.
• Lastly, if \[ B^2 - 4AC > 0 \], the result is a hyperbola.

In practice, calculating the discriminant offers a quick way to determine the shape of the graph without plotting points. As shown in the step-by-step solution, the negative discriminant led to identifying the equation as an ellipse.

Second-Degree Equations

Second-degree equations, or quadratic equations, form the basis for understanding conic sections. These equations typically appear as \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
and include a mix of squared and linear terms.
The types of conic sections they represent—parabolas, ellipses, and hyperbolas—depend largely on the interactions between coefficients A, B, and C as seen through the lens of the discriminant.

• The cross term \[ Bxy \] is important in understanding the tilt of the conic section.
• Completing the square or using rotations may convert these equations into their standard forms, making it easier to graphically analyze or identify the shapes.

Mastering second-degree equations means comprehending their role in diverse applications, from physics to engineering, where they model everything from freefall trajectories to electrical circuits.