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Maurinko [17]
3 years ago
15

Write the standard form of the equation of the circle with the given characteristics. Center: (3, 5); Solution point: (−2, 17)

Mathematics
1 answer:
Angelina_Jolie [31]3 years ago
3 0

\text{we know that the general equation of the circle is }\\
\\
(x-h)^2+(y-k)^2=r^2\\
\\
\text{where, (h,k) is the center of the circle and r is the radius.}\\
\\
\text{Given that the centered of the circle is at  }(h,k)=(3,5)\\
\text{and a solution point on circle is (-2,17).}\\
\\
\text{so the radius of the circle will be the distance between center and the }

\text{solution point. so using the distance formula, we have}\\
\\
\text{Radius, }r=\sqrt{(-2-3)^2+(17-5)^2}\\
\\
\Rightarrow r=\sqrt{25+144}\\
\\
\Rightarrow r=\sqrt{169}\\
\\
\Rightarrow r=13\\
\\
\text{Hence using the standard for of circle, equation of circle with center}\\
(h,k)=(3,5) \text{ and radius r=13 is}\\
\\
(x-3)^2+(y-5)^2=(13)^2\\
\\
\Rightarrow (x-3)^2+(y-5)^2=169

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(f – g) (-2) means the same as subtracting f(2) and g(2). Since we are given f(x) and g(x), we can use them to solve. There are two ways to solve. One is to find f(2) and g(2), and then subtract them. Another way is to do (f-g)(x), then plug in x=2. I will show both methods.

Method 1

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f(2)=3(4)+1                                  [multiply]

f(2)=12+1                                     [add]

f(2)=13

g(2)=1-(2)                                    [subtract]

g(2)=-1                  

(f-g)(2)=13-(-1)                            [subtract f(2) and g(2)]

(f-g)(2)=14

--------------------------------------------------------------------------------------------------------

Method 2

(f-g)(x)=3x²+1-(1-x)                     [distribute -1]

(f-g)(x)=3x²+1-1+x                      [combine like terms]

(f-g)(x)=3x²+x

(f-g)(2)=3(2)²+2                         [plug in x=2, exponent]

(f-g)(2)=3(4)+2                           [multiply]

(f-g)(2)=12+2                             [add]

(f-g)(2)=14

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Now, we know that (f-g)(2)=14. We confirmed this with both methods.

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Reasons:

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