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Chapter 5: Problem 62

The points of intersection of the graphs of \(x y=20\) and \(x^{2}+y^{2}=41\) are joined to form a rectangle. Find the area of the rectangle.

Short Answer

Expert verified
The area of the rectangle is \(\sqrt{82}\) square units.

Step by step solution

01

Find the intersection points

First, find the points of intersection between the two curves. Convert the equation \(xy = 20\) to \(y=20/x\). Then, substitute \(y = 20/x\) in the equation of the ellipse: \[(x^{2} + (20/x)^{2} = 41.\] This simplifies to a quadratic equation \(x^{4}-41x^2+400=0\). Solve this quadratic equation to get the x-coordinates of the intersection points.
02

Solve the quadratic equation

Now solve the quadratic equation using the quadratic formula: \[x = \frac{-b 卤 \sqrt{b^{2} - 4ac}}{2a}\]. This gives \(x^2 = 16\) and \(x^2 = 25\), yielding four possible x-coordinates for the intersection points: \(x = 卤4\) and \(x = 卤5\).
03

Find the y-coordinates of intersection points

Now, we substitute these values of x back into the equation \(y=20/x\) to calculate the respective y-coordinates. For \(x=卤4\), we get \(y=卤5\), and for \(x=卤5\), we get \(y=卤4\). Therefore, the points of intersection of the two curves are (4,5), (-4, -5), (5,4), and (-5, -4). These are the vertices of the rectangle.
04

Calculate the area of the rectangle

The two points (4,5) and (5,4) form a diagonal pair of vertices of the rectangle, with the distance between them as the length of the rectangle's diagonal. It can be calculated using Pythagoras Theorem as \[\sqrt{(5 - 4)^{2} + 5^{2}} = \sqrt{2}\]. Similarly, the points (4,5) and (-4,-5) form another pair of diagonal vertices. The distance between them, which is \[\sqrt{(5 + 5)^{2} + (4 + 4)^{2}} = 2 \sqrt{41}\], is the longer diagonal. Thus, the area of the rectangle is half the product of the lengths of the diagonals: \[\frac{1}{2} * \sqrt{2} * 2\sqrt{41} = \sqrt{82}\] square units.

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra and appear in various forms, usually written as ax^2 + bx + c = 0, where a, b, and c are coefficients, and x represents the variable or unknown. Solving these equations is a key skill for students, often utilizing methods such as factoring, completing the square, or applying the quadratic formula, which is \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \].

In the context of finding the intersection points of the graphs described in the exercise, transforming the condition of intersection into a quadratic equation allows for determining the precise x-coordinates of these points. After finding these x-values, which in this case are the square roots of 16 and 25, we then obtain the corresponding y-values, which completes the solution to the first step towards understanding the rectangle formation.
Pythagorean Theorem
The Pythagorean Theorem is an essential principle in geometry, particularly for right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, the theorem is expressed as \[ c^{2} = a^{2} + b^{2} \], where c is the hypotenuse, while a and b are the other two sides of the triangle.

In the textbook exercise, students must use the Pythagorean Theorem to determine the length of the diagonals of the rectangle, created by intersecting graphs. Understanding the application of this theorem is critical to accurately calculating the rectangle's area using the lengths of these diagonals as outlined in the solution steps.
Rectangular Area Calculation
The area of a rectangle is calculated by multiplying its length by its width. However, if the length and width are not directly known, there are alternative methods to find the area. For instance, when given the lengths of the diagonals of a rectangle, the area can be determined by calculating half the product of the diagonals' lengths. This method is derived from the properties of a right-angled triangle and the Pythagorean Theorem, which is used to calculate the length of the diagonals initially.

In the given exercise, once the length of both diagonals is determined, the area is easily computed. Remembering that the diagonals of a rectangle are congruent and that the rectangle can be divided into two right-angled triangles by its diagonal, it becomes a relevant and practical geometry exercise for students to test their understanding of these concepts.

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Most popular questions from this chapter

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\) $$ \begin{array}{l} a_{1} x+b_{1} y=c_{1} \\ a_{2} x+b_{2} y=c_{2} \end{array} $$

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