Problem 16 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}+12 x y+12 y^{2}+435 x-9 y+72=0\)

Step-by-Step Solution

Answer

The equation represents a parabola as the discriminant is zero.

Step 1: Identifying coefficients A, B, and C

First, let's identify the coefficients A, B, and C from the given equation. The equation is in the form of quadratic in the variables x and y:\[3x^2 + 12xy + 12y^2 + 435x - 9y + 72 = 0\]For a conic section given by \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the coefficients are:\( A = 3 \), \( B = 12 \), \( C = 12 \).

Step 2: Calculating the Discriminant

The discriminant for conic sections is calculated by the formula \( B^2 - 4AC \). Let's substitute the values of A, B, and C:\[B^2 - 4AC = 12^2 - 4 \times 3 \times 12 \]Calculate:\[144 - 144 = 0\]

Step 3: Interpreting the Discriminant

Now, examine the value of the discriminant:- If \( B^2 - 4AC > 0 \), the conic is a hyperbola.- If \( B^2 - 4AC = 0 \), the conic is a parabola.- If \( B^2 - 4AC < 0 \), the conic is an ellipse.Since \( B^2 - 4AC = 0 \), the equation represents a parabola.

Key Concepts

Discriminant

The discriminant is a special number that helps us find out what kind of shape a quadratic equation represents. It's very helpful for identifying conic sections like parabolas, ellipses, and hyperbolas.

To calculate the discriminant for conic sections, we use the formula:

• When the equation is in the form of: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is computed as: \(B^2 - 4AC\).

The value of this discriminant tells us about the curve's nature:

• If \(B^2 - 4AC > 0\), the equation represents a hyperbola.
• If \(B^2 - 4AC = 0\), the equation represents a parabola.
• If \(B^2 - 4AC < 0\), the equation represents an ellipse.

Knowing how to calculate the discriminant and interpret its value is key when studying conic sections. It helps you quickly determine the geometric shape of an equation.

Parabolas

A parabola is a U-shaped curve that can open upwards, downwards, leftward, or rightward. In the context of conic sections, parabolas are unique because they occur when the discriminant is zero.

Parabolas have the general equation:

• \(y^2 = 4ax\)
• \(x^2 = 4ay\)

Some key characteristics include:

• Vertex: The point where the parabola changes direction.
• Axis of symmetry: A line through the vertex, dividing the parabola into mirror images.
• Focus: A point inside the parabola, ensuring each point on the curve is equidistant from the focus and the directrix.
• Directrix: A straight line outside the parabola that helps define its shape along with the focus.

Interpreting the equation using the parabola's standard form helps in sketching and understanding its structure easily.

Ellipses

An ellipse looks like an elongated circle, or an oval. If the discriminant is less than zero, the equation represents an ellipse. An ellipse has two axes, a longer one called the major axis and a shorter one called the minor axis.

The standard form of an ellipse is:

• \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) if the major axis is along the x-axis.
• \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\) if the major axis is along the y-axis.

Unique qualities of ellipses include:

• Center: The midpoint of both axes.
• Foci: Two special points on the major axis such that the sum of distances from any point on the ellipse to the foci is constant.
• Vertices: The end points of the major axis.
• Co-vertices: The end points of the minor axis.

Ellipses have rich applications like planetary orbits and lenses, which makes understanding their properties very beneficial.

Hyperbolas

Hyperbolas consist of two separate curves, known as branches, which are mirror images of each other. They form when the discriminant is greater than zero. Unlike ellipses and parabolas, hyperbolas have a saddle shape that opens either horizontally or vertically.

Equations for hyperbolas look like:

• \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for horizontal hyperbolas.
• \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for vertical hyperbolas.

Characteristics of hyperbolas include:

• Center: The point equidistant from both branches' vertices.
• Vertices: Points where the branches are closest together.
• Asymptotes: Lines that the hyperbola approaches but never touches, helping define its direction.
• Conjugate axis: Perpendicular to the transverse axis, linking the co-vertices.
• Transverse axis: Contains both vertices and the center.

Understanding hyperbolas can help comprehend real-world phenomena like radio waves and the design of telescopes.