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Chapter 13: Problem 78

Find an equation of a circle satisfying the given conditions. Center \((-3,5)\) with a circumference of \(8 \pi\) units

Short Answer

Expert verified
(x+3)^2 + (y-5)^2 = 16

Step by step solution

01

Recall the formula for the circumference of a circle

The formula for the circumference of a circle is given by \[C = 2 \pi r\] where \(r\) is the radius. We are given that the circumference \(C\) is \(8 \pi\) units.
02

Solve for the radius

Using the circumference formula, set up the equation: \[8 \pi = 2 \pi r\] Divide both sides by \(2 \pi\) to solve for \(r\): \[r = \frac{8 \pi}{2 \pi} = 4\]
03

Recall the standard equation of a circle

The standard equation of a circle with center at \((h, k)\) and radius \(r\) is \[(x-h)^2 + (y-k)^2 = r^2\]
04

Substitute the center and radius

Given the center \((-3,5)\) and radius \(4\), substitute these values into the standard equation:\[(x+3)^2 + (y-5)^2 = 4^2\]
05

Simplify the equation

Simplify the equation to get the final form:\[(x+3)^2 + (y-5)^2 = 16\]

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Key Concepts

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

circumference of a circle
The circumference of a circle is the distance around the edge or the perimeter of the circle. It's like the circle's 'fence.' The formula to calculate the circumference is given by \(C = 2 \pi r\), where \(C\) represents the circumference and \(r\) is the radius of the circle. The symbol \(\pi\) (Pi) is a constant, approximately equal to 3.14159, but it is usually left as \(\pi\) in equations to maintain precision.
  • If you know the radius, you can find the circumference by doubling the radius and multiplying by \(\pi\).
  • In the original exercise, given the circumference \(8 \pi\), we can rearrange the formula to solve for radius.
This involves dividing the given circumference by \2 \pi\, which yields \(r = 4\). Hence, the radius of our circle is 4 units. Simple, right?
radius of a circle
The radius of a circle is the distance from the center of the circle to any point on its circumference. It is a constant distance for any point on the edge of the circle. In formulas, the radius is often denoted by \r\.

The radius connects crucially to other circle properties:
  • It is half the diameter of the circle.
  • The radius is essential for finding the circumference using the formula \(C = 2 \pi r\).
  • It is also vital for the area of the circle, calculated using \(A = \pi r^2\).
In our given exercise, the radius was calculated from the circumference \(8 \pi\) by the equation, resulting in \r=4\ units. This vital information feeds directly into forming the actual equation of the circle.
standard form of a circle equation
The standard form of a circle's equation is used to easily identify both the center and the radius of a circle. The standard equation is written as:
i \( (x-h)^2 + (y-k)^2 = r^2 \)
Here, \( (h,k) \) represent the coordinates of the circle's center and \( r \) is the radius.
  • The term \( (x-h) \) shifts the circle horizontally by \h\ units.
  • The term \( (y-k) \) shifts the circle vertically by \( k \) units.
  • The radius squared \( r^2 \) is on the right side of the equation.
In this exercise, given the center (-3, 5) and radius \4\, we substitute these values into the equation, yielding:
i \ (x+3)^2 + (y-5)^2 = 4^2 = 16 \.
This is the simplified standard form of the circle's equation related to the provided data.

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Most popular questions from this chapter

Evaluate each of the following. \(\frac{(-1)^{k}}{k-5},\) for \(k=10\)

The diagonal of a Persian rug is \(25 \mathrm{ft}\). The area of the rug is \(300 \mathrm{ft}^{2} .\) Find the length and the width of the rug.

For the following equations of hyperbolas, complete the square, if necessary, and write in standard form. Find the center, the vertices, and the asymptotes. Then graph the hyperbola. \(4 y^{2}-25 x^{2}-8 y-100 x-196=0\)

Let \((-c, 0)\) and \((c, 0)\) be the foci of an ellipse. Any point \(P(x, y)\) is on the ellipse if the sum of the distances from the foci to \(P\) is some constant. Use \(2 a\) to represent this constant. A) Show that an equation for the ellipse is given by \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-c^{2}}=1\) B) Substitute \(b^{2}\) for \(a^{2}-c^{2}\) to get standard form.

Solve. \(x^{2}+\frac{25}{x^{2}}=26\)
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