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Chapter 1: Problem 90

Find an equation of the circle that satisfies the given conditions. Center \((-1,-4) ;\) radius 8

Short Answer

Expert verified
The equation of the circle is \((x + 1)^2 + (y + 4)^2 = 64\).

Step by step solution

01

Recall the Standard Form of a Circle's Equation

The standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius of the circle.
02

Substitute the Center Coordinates

Substitute \(h = -1\) and \(k = -4\) into the standard form equation. This gives \((x + 1)^2 + (y + 4)^2 = r^2\).
03

Substitute the Radius

The given radius is 8, so substitute \(r = 8\). This results in \((x + 1)^2 + (y + 4)^2 = 8^2\).
04

Simplify the Equation

Calculate \(8^2\) to get 64. Thus, the equation of the circle is \((x + 1)^2 + (y + 4)^2 = 64\).

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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
When working with circles in mathematics, it's crucial to understand their equation representation. The standard form of a circle's equation is given by \[(x - h)^2 + (y - k)^2 = r^2,\]where:
  • \((h, k)\) represents the center of the circle.
  • \(r\) indicates the radius of the circle.
This form is quite powerful because it immediately shows the critical properties of a circle: its position and size.
By inserting the correct values for \(h\), \(k\), and \(r\), you can model any circle in a two-dimensional plane. This framework is widely used in geometry and various applications that involve circles.
Circle Center and Radius
The center and radius of a circle are paramount to understanding its equation and structure. The center, defined by \((h, k)\), determines the circle's location on a Cartesian coordinate system. It's like pinpointing the exact spot where you want your circle to sit.
The radius \(r\), on the other hand, decides how large or small the circle will be. It's the fixed distance from the center to any point on the circle's boundary.
  • A larger radius means a bigger circle.
  • A smaller radius signifies a tighter circle.
These parameters, center and radius, together create every possible circle that you can graph on a plane.
By plugging them into the standard form, they define not only the shape but also ensure that the circle is perfect without any elliptical distortions.
Circle Equation Derivation
Deriving the equation of a circle may sound complex, but it's straightforward when you follow certain steps. Knowing the circle's center and radius allows you to substitute these values into the standard form equation. For example, if a circle's center is \((-1, -4)\) and its radius is 8, you would proceed as follows:
  • Start with the standard form: \((x - h)^2 + (y - k)^2 = r^2\).
  • Substitute \(h = -1\) and \(k = -4\) into the equation:\((x + 1)^2 + (y + 4)^2 = r^2\).
  • Insert the given radius \(r = 8\) in the equation:\((x + 1)^2 + (y + 4)^2 = 8^2\).
  • Calculate and simplify \(8^2\) to get 64, which completes the equation:\((x + 1)^2 + (y + 4)^2 = 64\).
These steps not only produce the circle's exact equation but also reinforce understanding of how each component influences the result.
You end up with an equation that accurately represents the circle on a graph, encapsulating both its position and dimensions.

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Most popular questions from this chapter

Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0},\) where \(h\) is measured in feet. Use this information to solve the problem. A ball is dropped from the top of a building 96 ft tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level?

Simplify the expression. (a) \(\frac{w^{4 / 3} w^{2 / 3}}{w^{1 / 3}}\) (b) \(\frac{a^{5 / 4}\left(2 a^{3 / 4}\right)^{3}}{a^{1 / 4}}\)

Write the number indicated in each statement in scientific notation. (a) The distance from the earth to the sun is about 93 million miles. (b) The mass of an oxygen molecule is about 0.000000000000000000000053 g. (c) The mass of the earth is about \(5,970,000,000,000,000,000,000,000 \mathrm{kg}\)

Simplify the expression. (a) \(\left(8 a^{6} b^{3 / 2}\right)^{2 / 3}\) (b) \(\left(4 a^{6} b^{8}\right)^{3 / 2}\)

DISCOVER - PROVE: Relationship Between Solutions and Coefficients The Quadratic Formula gives us the solutions of a quadratic equation from its coefficients. We can also obtain the coefficients from the solutions. (a) Find the solutions of the equation \(x^{2}-9 x+20=0\) and show that the product of the solutions is the constant term 20 and the sum of the solutions is \(9,\) the negative of the coefficient of \(x\) (b) Show that the same relationship between solutions and coefficients holds for the following equations:$$ \begin{array}{l}x^{2}-2 x-8=0 \\\x^{2}+4 x+2=0\end{array}$$ (c) Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has solutions \(r_{1}\) and \(r_{2}\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)
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