/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Sketch the graph of the circle o... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 25

Sketch the graph of the circle or semicircle. $$x^{2}+y^{2}=11$$

Short Answer

Expert verified
Circle with center (0, 0) and radius \(\sqrt{11}\).

Step by step solution

01

Identify the equation type

The equation provided, \(x^2 + y^2 = 11\), represents a circle. This is because it matches the standard form equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
02

Determine the center

The circle equation \(x^2 + y^2 = 11\) does not have any \(h\) or \(k\) values, meaning \(h = 0\) and \(k = 0\). Therefore, the center of the circle is at the origin, which is the point \((0, 0)\).
03

Find the radius

In the equation \(x^2 + y^2 = 11\), the number on the right side, 11, is equal to \(r^2\). To find the radius \(r\), take the square root of 11. Thus, the radius \(r = \sqrt{11}\).
04

Sketch the circle

Draw a circle with the center at \((0, 0)\) and a radius of \(\sqrt{11}\). Use a compass or a digital tool to ensure the circle is accurately sized, touching the axes at points approximately \((\pm \sqrt{11}, 0)\) and \((0, \pm \sqrt{11})\).

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Key Concepts

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

Standard Form of a Circle
Equations of circles have a special format known as the "standard form of a circle". Understanding this form is crucial when dealing with problems involving circles in geometry.
In its simplest form, the standard equation is expressed as \((x - h)^2 + (y - k)^2 = r^2\).
  • \((h, k)\) are the coordinates of the circle's center.
  • \(r\) is the radius of the circle.
This structure helps in quickly identifying vital attributes of the circle, like its position and size, just from looking at the equation. If you see an equation such as \(x^2 + y^2 = 11\), it aligns perfectly with the standard form but with no shifts from zero, indicating the default center at the origin. The given number, 11, is equal to \(r^2\), hinting you must find the square root to solve for the radius.
Center of a Circle
When determining the center of a circle, the focus is on the values \((h, k)\) in the standard form equation \((x - h)^2 + (y - k)^2 = r^2\). These values tell you precisely where the center is situated in your coordinate plane.
If the circle's equation simplifies to \(x^2 + y^2 = r^2\), such as \(x^2 + y^2 = 11\), it suggests that the center is at \(0, 0\) because neither \(x\) nor \(y\) have constants that shift it away from zero.
  • Check the values of \(h\) and \(k\); the absence of additional constants means both are zero.
  • The origin, \((0, 0)\), is the central point of this particular circle.
Thus, recognizing that there's no horizontal or vertical shift confirms that calculations revolve around the coordinate origin.
Radius of a Circle
The radius of a circle is the constant distance from its center to any point on its circumference. In circle equations, the radius is denoted as \(r\) and is represented in the standard form equation as \(r^2\).
Here's how you'd solve for \(r\):
If given \(x^2 + y^2 = 11\), you have \(r^2 = 11\).
  • To find \(r\), take the square root of both sides.
  • Thus, \(r = \sqrt{11}\).
This tells us that the radius is not a whole number but the square root of 11, about 3.32 when approximated. This length determines how far the circle stretches from its center across the plane.

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Most popular questions from this chapter

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