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Chapter 7: Problem 140

Find the equation of a circle satisfying the given conditions. Center: (5,-2)\(;\) radius: 4

Short Answer

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

Step by step solution

01

Know the standard equation of a circle

The standard equation of a circle with center \(h, k\) and radius \R\ is given by \((x - h)^2 + (y - k)^2 = R^2\).
02

Substitute the center coordinates

Substitute \(h = 5\) and \(k = -2\) into the standard equation of a circle: \((x - 5)^2 + (y + 2)^2 = R^2\).
03

Substitute the radius

The given radius is 4. Substituting \R = 4\ into the equation gives: \((x - 5)^2 + (y + 2)^2 = 4^2\).
04

Simplify the equation

Simplify \4^2\ which is 16. Hence, the equation becomes \((x - 5)^2 + (y + 2)^2 = 16\).

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Key Concepts

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

standard equation of a circle
When dealing with circles in coordinate geometry, it's essential to know their standard equation. This can be represented as \((x - h)^2 + (y - k)^2 = R^2\), where \h\ and \k\ are the coordinates of the center of the circle, and \R\ is the radius.
This equation expresses that all points \(x, y\) on the circle are at a constant distance \R\ from the center point \(h, k\) . Understanding this foundational formula is crucial for solving any circle-related problems efficiently.
coordinates of the center
The coordinates of the center of a circle are given by the values of \h\ and \k\ in the standard equation. For example, in our exercise, the center is given as \(5, -2\).
To incorporate this into our formula, we substitute these values for \h\ and \k\. Hence, the standard equation \((x - h)^2 + (y - k)^2 = R^2\) becomes \((x - 5)^2 + (y + 2)^2 = R^2\).
This step translates the center's coordinates into the mathematical framework of the circle's equation.
radius substitution
The radius of the circle operates as the constant \R\ in the standard circle equation.
Substituting this value is simple yet critical. For our specific problem, \R\ is given as 4.
By plugging in this value, our equation \((x - 5)^2 + (y + 2)^2 = R^2\) turns into \((x - 5)^2 + (y + 2)^2 = 4^2\).
This step ensures that the equation aligns precisely with the circle's designated size, preserving its geometric properties.
simplification of equations
The final step involves simplifying the equation for easier interpretation and use.
Here, we simplify \4^2\, which equals 16.
Thus, the equation \((x - 5)^2 + (y + 2)^2 = 4^2\) simplifies to \((x - 5)^2 + (y + 2)^2 = 16\).
This final form of the circle's equation is more concise and ready for application in other calculations or problems.
Remember, simplification not only makes the equation cleaner, but also helps in identifying and solving further mathematical tasks involving the circle.

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