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Chapter 11: Problem 19

Write the standard form of the equation of the circle with the given radius and center \((0,0) .\) $$ \text { Radius: } \sqrt{2} $$

Short Answer

Expert verified
The standard form of the equation of the circle is \[ x^2 + y^2 = 2 \]

Step by step solution

01

Identify the standard form of a circle's equation

The standard form of a circle's equation with center \(h,k\) and radius r is \[ (x-h)^2 + (y-k)^2 = r^2 \]
02

Substitute the center coordinates

Since the center of the circle is at \(0,0\), substitute \(h = 0\) and \(k = 0\) into the equation. It simplifies to: \[ x^2 + y^2 = r^2 \]
03

Substitute the radius

Given the radius is \sqrt{2}\, substitute \(r = \sqrt{2}\) into the equation. Thus, \(r^2 = (\sqrt{2})^2 = 2\).\
04

Write the final equation

Combining the steps, the equation of the circle is: \[ x^2 + y^2 = 2 \]

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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 Equation
The standard form of the equation of a circle is a fundamental concept in geometry. It provides a way to represent a circle mathematically, describing its location and size.
The general equation is written as: \[ (x-h)^2 + (y-k)^2 = r^2 \]
Here:
  • \( (h, k) \) represents the coordinates of the center of the circle.
  • \( r \) is the radius of the circle.
This form makes it easy to identify both the center and radius, which are crucial for graphing and solving related problems.
Circle Equation
A circle's equation showcases how its various elements interact with one another. The components, such as the radius and center, are all integrated into the equation:
For a circle centered at \((h, k)\) with radius \(r\), the equation is: \[ (x-h)^2 + (y-k)^2 = r^2 \]
Breaking this down further:
  • \((x-h)^2\) shifts the circle horizontally by \(h\).
  • \((y-k)^2\) shifts the circle vertically by \(k\).
  • \( r^2 \) defines the stretch of the circle, determining its size based on how large the radius is.
This encapsulates everything needed to understand a circle's properties on the Cartesian plane.
Substitution Method
The substitution method is a straightforward technique to plug in known values into an equation. It's applied here to find the specific equation of a circle given its radius and center.
Steps involved in this context include:
  1. Identify the standard form equation: \[ (x-h)^2 + (y-k)^2 = r^2 \]
  2. Substitute the given center coordinates into the equation. For a center at \((0,0)\), we would substitute \( h=0 \) and \( k=0 \), simplifying to \[ x^2 + y^2 = r^2 \]
  3. Substitute the given radius into the equation. For a radius of \( \sqrt{2} \), we substitute \( r = \sqrt{2}\), so \[ r^2 = (\sqrt{2})^2 = 2 \]
By following these steps, you determine the specific form of the circle's equation.
Radius and Center of a Circle
Understanding the radius and center is crucial for working with circles. These components define the size and location:
  • The center is the fixed point from which every point on the circle is equidistant. In a circle with the center at \((0,0)\), this means all points are equally distant from this origin point.
  • The radius is the distance from the center to any point on the circle. It helps determine the size of the circle. For example, a radius of \(\sqrt{2}\) means every point on the edge of the circle is a distance of \( \sqrt{2} \) units from the center.
Knowing these components lets you construct and analyze the circle's geometric properties more accurately.

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Most popular questions from this chapter

(a) write the equation in standard form and (b) graph. $$ 9 x^{2}-4 y^{2}-18 x+8 y-31=0 $$

Graph the equation. $$ x^{2}+y^{2}=64 $$

Graph the equation. $$ x=4(y+1)^{2}-4 $$

Identify the type of graph. (a) \(x=-2 y^{2}-12 y-16\) (b) \(x^{2}+y^{2}=9\) (c)\(16 x^{2}-4 y^{2}+64 x-24 y-36=0\) (d)\(16 x^{2}+36 y^{2}=576\)

Solve the system of equations by using graphing. $$ \left\\{\begin{array}{l} y=\frac{3}{2} x+3 \\ y=-x^{2}+2 \end{array}\right. $$
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