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Chapter 1: Problem 74

Write the standard form of the equation of the circle with the given characteristics. Center: \((3,-2) ;\) Solution point: \((-1,1)\)

Short Answer

Expert verified
The standard form equation of the circle is \((x-3)^2 + (y+2)^2 = 25\).

Step by step solution

01

Identify Center and Point on Circle

The center of circle is given as \((3,-2)\), represented as (h, k). Another point \((-1,1)\) that lies on the circle is also provided.
02

Calculate Radius Using the Distance Formula

Use the distance formula to find the radius (r) of the circle, which is the distance between the center \((3,-2)\) and the given point \((-1,1)\). The formula for distance between two points (\(x_1, y_1)\) and (\(x_2,y_2)\) is \(d = \sqrt {[(x_2 - x_1)^2 + (y_2 - y_1)^2]}\), here \(x_1=3, y_1=-2, x_2=-1, y_2=1\) which gives rise to \(d = \sqrt{ [(-1-3)^2 + (1-(-2))^2]}\) on calculating it results in \(d = \sqrt{16+9}\), which is \(d = 5\).
03

Write Equation of Circle in Standard Form

Now that all components are known, substitute h=3, k=-2 and r=5 into the standard circle equation \( (x-h)^2 + (y-k)^2 = r^2 \). This provides the equation \((x-3)^2 + (y+2)^2 = 25\).

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Most popular questions from this chapter

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