Problem 7 - Algebra 2

Question

When an airplane flies faster than the speed of sound, it produces a shock wave in the shape of a cone. Suppose the shock wave generated by a jet intersects the ground in a curve that can be modeled by the equation \(x^{2}-14 x+4=9 y^{2}-36 y .\) Identify the shape of the curve.

Step-by-Step Solution

Answer

The curve is a hyperbola.

Step 1: Rewrite Equation

First, rewrite the given equation, which is \( x^2 - 14x + 4 = 9y^2 - 36y \), in a standard form for conic sections. To do this, move all terms to one side of the equation, resulting in \( x^2 - 14x + 4 - 9y^2 + 36y = 0 \).

Step 2: Complete the Square for x-terms

Focus on completing the square for the x-terms. The quadratic expression is \( x^2 - 14x \). Take half of the coefficient of x, which is -7, square it to get 49, and add/subtract 49 to maintain equality: \( x^2 - 14x = (x - 7)^2 - 49 \).

Step 3: Complete the Square for y-terms

Next, complete the square for the y-terms \( -9y^2 + 36y \). Factor out -9: \( -9(y^2 - 4y) \). Half of the coefficient of y is 2, squaring gives 4. Add/subtract this inside the bracket: \( -9((y - 2)^2 - 4) = -9(y - 2)^2 + 36 \).

Step 4: Simplify and Analyze

Substitute the completed squares back into the equation: \( (x - 7)^2 - 49 - 9(y - 2)^2 + 36 = 0 \). Simplify: \( (x - 7)^2 - 9(y - 2)^2 = 13 \). This equation resembles \( (x-h)^2/b^2 - (y-k)^2/a^2 = 1 \), which denotes a hyperbola.

Key Concepts

Conic Sections

Conic sections are fundamental shapes formed by slicing a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas. Each shape has its own unique set of properties and equations.

• **Circle:** The simplest conic section, a circle is formed when the plane cuts the cone parallel to its base. Its equation is of the form \(x^2 + y^2 = r^2\).
• **Ellipse:** An ellipse appears when the plane cuts through the cone at an angle. The standard form of its equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
• **Parabola:** A parabola arises when the plane is parallel to the cone's side. It has the equation \(y = ax^2 + bx + c\).
• **Hyperbola:** A hyperbola looks like two mirrored curves and is formed when the plane cuts both halves of the cone. Its standard equation is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).

In the exercise, the shock wave from the jet forms a curve called a hyperbola upon intersecting the ground. This shape is determined by analyzing the equation using its geometric properties.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a more recognizable form, making it easier to analyze and solve.
This process is especially useful in conic sections for identifying the vertex or center of the shapes.
To complete the square for a quadratic equation like \(ax^2 + bx + c\):

• Take the coefficient of \(x\), divide it by 2, then square it. Add and subtract this square within the equation.
• This transforms the quadratic into a perfect square trinomial, \((x - p)^2\), making it easier to handle mathematically.
• Don't forget to account for the constant adjustments made when both adding and subtracting the squared term.

In the given problem, completing the square is applied separately to both \(x\) and \(y\) terms. This helps to rewrite the equation into a standard form that clearly shows the hyperbola's properties.

Standard Form Equations

Standard form equations for conic sections are crucial for classifying and analyzing these curves.
Each type of conic section has a specific standard form that reveals its essential properties and orientation.
For hyperbolas, the standard form is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\). This indicates a hyperbola centered at \((h,k)\) with its transverse and conjugate axes lengths determined by \(a\) and \(b\).
Other forms include standard equations for ellipses and parabolas, each tailored to highlight defining characteristics like the center or vertex.
Bringing the equation into standard form helps to visually identify and distinguish between these conic sections. This step is fundamental when solving or sketching conic sections in geometry and algebra.
In the exercise, converting the given equation to its standard hyperbolic form confirms the curve's identity as a hyperbola, helping us understand its structure better.