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Chapter 13: Problem 27

A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary and in the interior of the region bounded by the positive \(x\) -axis, the positive y-axis, the graph of \(x^{2}+y^{2}=25,\) and the line passing through \((-3,0)\) and \((0,2) ?\)

Short Answer

Expert verified
In total, there are 18 lattice points, 12 within the region and 6 on the boundary.

Step by step solution

01

Plot the Region

Firstly, create an imaginary graph on which the region given by \(x^{2}+y^{2}=25\) is plotted. This is a circle with radius 5 centered at the origin. Include the positive x-axis and the positive y-axis. Then, plot the line given by the points \((-3,0)\) and \((0,2)\). This line would bound the area of interest to a quarter of the circle.
02

Identify the Lattice Points on the Boundary

Secondly, the points on the curve \(x^{2}+y^{2}=25\) and the line running between \((-3,0)\) and \((0,2)\) where both x and y are integers are lattice points on the boundary. These points are \((-3,0),(0,2),(0,0),(5,0),(3,4),(0,5)\) for a total of 6 points.
03

Identify the Lattice Points in the Interior

Lastly, consider the lattice points inside the bounded region. Here, you are looking for points where both x and y are integers and they are inside the quadrilateral region. These points are \((1,0),(2,0),(3,0),(4,0),(0,1),(1,1),(2,1),(3,1),(0,3),(1,2),(2,2),(0,4)\) for a total of 12 points.

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Key Concepts

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

Coordinate Geometry
Coordinate Geometry is a branch of geometry that deals with the study of geometric figures using a coordinate system. This approach helps in analyzing shapes through algebraic equations, making it easier to understand various geometric properties. It involves:
  • Defining the position of points using ordered pairs known as coordinates.
  • Using a coordinate grid, usually with a horizontal x-axis and a vertical y-axis.
  • Applying algebraic methods to solve geometric problems, such as finding distances or midpoints.
In this exercise, coordinate geometry allows us to visualize the circle and the line segment on a coordinate plane, helping us identify the lattice points within these boundaries. The circle has its center at the origin, and its properties are easily evaluated using the equation provided.
Equation of Circle
The Equation of a Circle in a coordinate plane is given by the formula \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle, and \((0,0)\) is the center of the circle in the case of a circle centered at the origin. Here:
  • The equation \(x^2 + y^2 = 25\) represents a circle with a radius of 5 units centered at the origin.
  • This can be visualized by considering all points that satisfy this equation lie on the circumference of the circle.
  • The circle is bounded by the coordinate axes, creating a defined region we are analyzing.
Recognizing this circle's boundaries is crucial in determining which lattice points lie on the circle's circumference or inside the circle.
Integer Coordinates
Integer Coordinates refer to points on the coordinate plane where both the \(x\) and \(y\) values are integers. These points are commonly known as lattice points.
  • Lattice points are useful for mathematical problems involving counting or geometry.
  • In this scenario, lattice points can help determine the precise numbers of points lying on the boundary and within the interior of the given region.
  • These points provide specific reference positions where the coordinates \((x, y)\) are whole numbers, simplifying calculations.
In our problem, identifying lattice points involves finding integer solutions to equations describing the circle and straight line, depicting both those inside and on the boundary of the defined region.
Line Equation
The Line Equation is a fundamental concept in coordinate geometry, often expressed in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.For a line between two points, like \((-3,0)\) and \((0,2)\), calculate the slope as: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{0 - (-3)} = \frac{2}{3}\).Hence, the equation becomes: \(y = \frac{2}{3}x + 2\). Here's the role of the line equation in our task:
  • It helps delineate the upper boundary of our region, determining which lattice points fall inside or outside.
  • It assists in identifying integer coordinate points that align precisely on the line, and those within the defined region.
  • A careful evaluation of the line helps accurately count boundary and interior points within the sector defined by the axes, circle, and line.
Understanding the line equation empowers you to distinguish precisely how geometric shapes interact with an algebraic perspective.

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Most popular questions from this chapter

Consider the equations of three lines: \(y=x+4\) \(y=3 x \quad y=2 x-2\) If two of the three lines are selected at random, what is the probability that both contain the point \((2,6) ?\)

Is \((6,8)\) on the graph of \(x^{2}+y^{2}=100 ?\)

\(\text { In } \triangle \mathrm{AOB}, \mathrm{A}=(6,0), \mathrm{B}=(0,8), \text { and } \mathrm{O}=(0,0)\) a. Find, to the nearest tenth, the volume of the solid formed by rotating the triangle about \(\overline{\mathrm{OA}}\). b. Find, to the nearest tenth, the volume of the solid formed by rotating the triangle about \(\overline{\mathrm{OB}}\). c. Find, to the nearest tenth, the volume and the total surface area of the solid formed by rotating the triangle about \(\overline{\mathrm{AB}}\).

Find the distance from \(\mathrm{P}=(3,4,12)\) to the origin.

Determine the equation of each circle. a. The center is the origin, and the circle passes through \((0,-5)\) b. The endpoints of a diameter are \((-2,1)\) and \((8,25)\) c. The center is \((-1,7),\) and the circle passes through the origin. d. The center is \((2,-3),\) and the circle passes through \((3,0)\)
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