Question

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the equation of a circle is given below.
$x^2 + (y - 2.25)^2 = \frac{196}{169}$
what is its center?
( , )
what is its radius?
if necessary, round your answer to two decimal places.
units
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Explanation:

Step1: Recall the standard form of a circle's equation

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

Step2: Identify the center

For the equation \(x^{2}+(y - 2.25)^{2}=\frac{196}{169}\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h = 0\) (since \(x=x-0\)) and \(k = 2.25\).

Step3: Calculate the radius

We know that \(r^{2}=\frac{196}{169}\). Taking the square root of both sides, \(r=\sqrt{\frac{196}{169}}\). Since \(\sqrt{\frac{196}{169}}=\frac{\sqrt{196}}{\sqrt{169}}\), and \(\sqrt{196} = 14\), \(\sqrt{169}=13\), so \(r=\frac{14}{13}\approx1.08\).

Answer:

Center: \((0,2.25)\)
Radius: \(1.08\) units