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Chapter 14: Problem 10

Determine the locus of the equation \(b^{2}-2 x^{2}=2 x y+y^{2}\) (Hint \(:\) Add \(x^{2}\) to both sides.)

Short Answer

Expert verified
Question: Find the locus of the equation \(b^2 - 2x^2 = 2xy + y^2\). Answer: The locus of the given equation represents a hyperbola.

Step by step solution

01

Add \(x^2\) on both sides of the given equation

Add \(x^2\) to both sides of the equation: \((b^2 - 2x^2) + x^2 = (2xy + y^2) + x^2\) Now the equation becomes: \(b^2 - x^2 = x^2 + 2xy + y^2\)
02

Complete the square on the left and right sides

Observe that we can write the left and right sides of the equation as the difference and sum of squares, respectively: \((b^2 - x^2) = (x^2 + 2xy + y^2)\) Written in terms of squares: \((b - x)(b + x) = (x + y)^2\)
03

Divide both sides by \((b+x)\)

Now, let's divide both sides of the equation by \((b + x)\): \(\frac{(b - x)(b + x)}{b + x} = \frac{(x + y)^2}{b + x}\) The \((b + x)\) terms cancel on the left side: \(b - x = \frac{(x + y)^2}{b + x}\)
04

Express y in terms of x and b

Move the left side of the equation to the right side: \(0 = \frac{(x + y)^2}{b + x} - (b - x)\) Now, multiply both sides by \((b+x)\), which gives us: \(0 = (x + y)^2 - (b - x)(b + x)\) Now, we have: \((b - x)(b + x) = (x + y)^2\) Notice that on the left side, we have the difference of two squares again: \((b^2 - x^2) = (x + y)^2\) The locus of this equation is a combination of the properties of a hyperbola and a parabola. We can determine that this locus represents a hyperbola, as it corresponds to the difference of two squares, and retains some properties of a parabola with the equation \((x + y)^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a mathematical technique used to rewrite quadratic equations in a way that reveals important features, like the vertex of a parabola or to solve equations involving conic sections. Let's break it down step by step.

  • First, isolate terms involving the square, if necessary. In our example, we added \(x^2\) to both sides of the equation to reorganize it.
  • Next, look at the terms you want to complete as a perfect square. For terms like \(x^2 + 2xy + y^2\), you can see these form the square of \((x+y)^2\).
  • Write these terms as squared expressions, such as \((x+y)^2\) on one side and \((b - x)(b + x)\) on the other, which can also reveal structure.
Understanding the process of completing the square is essential in transforming expressions into a form where more can be deduced about their geometric nature, such as in the case of conics like parabolas and hyperbolas. This brings us to the next concept on our list.
Difference of Squares
The difference of squares is an algebraic expression that can be factored into the form \(a^2 - b^2 = (a - b)(a + b)\). This factoring reveals two things multiplied together.

  • It's commonly used when dealing with expressions like \(b^2 - x^2\), which directly factors into \((b - x)(b + x)\).
  • This factorization helps simplify the equation, making it easier to solve or interpret geometrically.
  • Recognizing the difference of squares allows us to transition from seemingly complex expressions to simpler products.
Applying this to our problem helped in simplifying the equation and understanding the shape it represents. In particular, this kind of manipulation shows how the equation fits the structure of a hyperbola, as we'll see in the next section.
Hyperbola Equation
A hyperbola is a type of conic section defined by its characteristic formula which often involves a difference of squares. The standard form of a rectangular hyperbola looks like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

  • The equation given in the problem can be manipulated to show a hyperbolic structure, specifically by reorganizing and factorization.
  • The recognition of the form \((b^2 - x^2) = (x + y)^2\) gives insight into its hyperbolic nature.
  • Hyperbolas are significant because they represent loci of points satisfying a particular difference from the foci, which in terms of geometry, can model several scientific phenomena.
By understanding the manipulation of equations into a form that reveals these geometric shapes, one can gain deeper insights into the fundamental properties of hyperbolas, as demonstrated in the exercise.

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Most popular questions from this chapter

Show that the odds against at least one 1 appearing in a throw of three dice is \(125: 91\). (This answer was stated by Cardano.)

Show that de Witt's equation $$ y^{2}+\frac{2 b x y}{a}+2 c y=\frac{f x^{2}}{a}+e x+d $$ represents a hyperbola. (Use the substitution \(z=y+\frac{b}{a} x+\) \(c\) and show that this substitution, when combined with a substitution of the form \(x^{\prime}=\beta x\), converts the original oblique \(x-y\) coordinate system into a new \(x^{\prime}-z\) coordinate system based on perpendicular axes.) Sketch the curve.

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If two players play a game with two dice with the condition that the first player wins if the sum thrown is 7 , the second wins if the sum is 6 , and the stakes are split if there is any other sum, find the expectation (value of the chance) of each player.
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