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Chapter 13: Problem 11

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

Short Answer

Expert verified
Yes, the point (6,8) lies on the graph of \(x^{2}+y^{2}=100\).

Step by step solution

01

Understand the circle equation

In the equation \(x^{2}+y^{2}=100\), we have a circle centered at the origin (0,0) with a radius of \(\sqrt{100}=10.\) To check if a point (a,b) is on this circle, we substitute a for x and b for y in the equation. If the equation holds true, the point is located on the graph. If not, the point does not lie on the circle.
02

Substitute the coordinates

Substitute x = 6 and y = 8 into the equation. The equation then becomes \(6^{2} + 8^{2} = 100\). Calculate the left side.
03

Compare with the right side

Calculate the left side of the equation: \(6^{2} + 8^{2} = 36+64 =100\). Compare the calculated result with the radius squared (right side of the equation): 100. As they match, the coordinates (6,8) indeed lie on the circle.

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Key Concepts

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

Understanding the Radius of a Circle
In any circle equation, the radius plays a crucial role. The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center, and \(r\) is the radius. For our exercise, the center is \((0, 0)\), making the equation \(x^2 + y^2 = r^2\). The given equation is \(x^2 + y^2 = 100\), which means that \(r^2 = 100\). To find the radius \(r\), we simply take the square root: \(r = \sqrt{100} = 10\). Thus, the radius of our circle is 10 units. This value helps determine how far any point on the circle is from the center. Knowing the radius is essential when verifying whether a point lies on a specific circle.
Working with Coordinates
Coordinates \((x, y)\) are a way to precisely locate a point on the cartesian plane. Each coordinate pair gives you the horizontal \(x\) and vertical \(y\) positions. In this exercise, we test the point \((6, 8)\) to see if it falls on the circle with the equation \(x^2 + y^2 = 100\). The process involves substituting these coordinates into the circle equation.

  • Substitute \(x = 6\) and \(y = 8\).
  • Calculate the expression \(x^2 + y^2\) to check if it equals 100.
By substituting, we find \(6^2 + 8^2 = 36 + 64 = 100\). Since the results are equal, the coordinates \((6, 8)\) do lie on the circle. This straightforward substitution method is key to confirming a point's location in relation to a circle.
Interpreting the Graph of the Circle's Equation
The graph of a circle's equation provides a visual representation. It helps to better understand the relationship between the equation and the shape. Graphing \(x^2 + y^2 = 100\) will result in a perfectly round circle on a cartesian plane, centered at the origin \((0, 0)\) with a radius of 10 units.

To plot or interpret this graph:
  • The center of the circle is at \((0, 0)\).
  • The circle includes all points that are exactly 10 units away from the center.
  • You can use the equation to test specific coordinates to see if they lie on the circle, like checking \((6, 8)\).
An understanding of the graph also aids in visual assessments, making it easier to predict and check the possible location of points. This combination of visual and numerical checking ensures a robust understanding of the circle's equation and its properties.

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Most popular questions from this chapter

Consider the points \(A=(2,3,-5), B=(8,9,1),\) and \(\mathrm{C}=(3,17,1)\) a Find the midpoint of \(\overline{\mathrm{AB}}\). b Find, to the nearest tenth, the length of the median from C to \(\overline{\mathrm{AB}}\).

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 \((3,0)\)

Find the distance between the parallel lines corresponding to \(y=2 x+3\) and \(y=2 x+7 .\) (Hint: Start by choosing a convenient point on one of the lines.)

Find the points each pair of lines has in common. a. \(\left[\begin{array}{l}2 x+y=10 \\ 8 x+4 y=17\end{array}\right.\) b. \(\left[\begin{array}{l}y=4 x+1 \\ \text { The line to the right of the } y \text { -axis, } \\ \text { parallel to it, and } 4 \text { units from it }\end{array}\right.\)

Write (if possible, in point-slope form) an equation of the line a Containing \((2,1)\) and \((3,4)\) b Containing \((-6,3)\) and \((2,-1)\) c Containing \((1,5)\) and \((-3,5)\) d With an x-intercept of 2 and a slope of 7 e That has an x-intercept of 3 and passes through \((1,8)\) f That passes through \((-3,6)\) and \((-3,10)\) g That passes through \((8,7)\) and is perpendicular to the graph of \(3 y=-2 x+24\)
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