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Chapter 42: Problem 730

Write the equation of the family of all concentric circles whose common center is the point \((-3,5)\). Draw three members of the family, specifying the value of the parameter in each case.

Short Answer

Expert verified
The family of concentric circles with a common center at point \((-3,5)\) can be represented by the equation \((x+3)^2+(y-5)^2=r^2\), where \(r\) is the radius. Three members of the family can be obtained by choosing different radii: 1. For \(r_1=2\), the equation is \((x+3)^2+(y-5)^2=4\). 2. For \(r_2=4\), the equation is \((x+3)^2+(y-5)^2=16\). 3. For \(r_3=6\), the equation is \((x+3)^2+(y-5)^2=36\).

Step by step solution

01

General Equation of a Circle

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

Equation of Concentric Circles with Common Center \((-3,5)\)

Using the given center \((-3,5)\), the equation of the concentric circles becomes: \[(x + 3)^2+(y-5)^2 = r^2\] This represents the family of all concentric circles with the common center at point \((-3,5)\).
03

Three Members of the Family (Different Radii)

To represent three members of the family, we will choose three different values for the radius \(r\). Let's take \(r_1 = 2\), \(r_2 = 4\), and \(r_3 = 6\). Circle with \(r_1 = 2\): \[(x + 3)^2 + (y - 5)^2 = (2)^2\] \[(x + 3)^2 + (y - 5)^2 = 4\] Circle with \(r_2 = 4\): \[(x + 3)^2 + (y - 5)^2 = (4)^2\] \[(x + 3)^2 + (y - 5)^2 = 16\] Circle with \(r_3 = 6\): \[(x + 3)^2 + (y - 5)^2 = (6)^2\] \[(x + 3)^2 + (y - 5)^2 = 36\] These three equations represent three members of the family of concentric circles with a common center at point \((-3,5)\) and radii of \(2\), \(4\), and \(6\) respectively.

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