Questions: Determine the center and radius of the circle. Write your answers as si [ (x+frac43)^2+(y-frac15)^2=frac1681 ]

Solution Steps

To determine the center and radius of the circle given by the equation \(\left(x+\frac{4}{3}\right)^{2}+\left(y-\frac{1}{5}\right)^{2}=\frac{16}{81}\), we can compare it to the standard form of a circle's equation \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) is the center of the circle and \(r\) is the radius.

1. Identify the center \((h, k)\) by comparing the given equation to the standard form.
2. Determine the radius \(r\) by taking the square root of the right-hand side of the equation.

To determine the center of the circle, we compare the given equation \(\left(x+\frac{4}{3}\right)^{2}+\left(y-\frac{1}{5}\right)^{2}=\frac{16}{81}\) to the standard form of a circle's equation \((x-h)^2 + (y-k)^2 = r^2\).

From the given equation: \[ (x + \frac{4}{3})^2 + (y - \frac{1}{5})^2 = \frac{16}{81} \]

We can see that: \[ h = -\frac{4}{3} \quad \text{and} \quad k = \frac{1}{5} \]

Thus, the center of the circle is: \[ \left( -\frac{4}{3}, \frac{1}{5} \right) \]

To find the radius, we take the square root of the right-hand side of the equation.

Given: \[ r^2 = \frac{16}{81} \]

Taking the square root of both sides: \[ r = \sqrt{\frac{16}{81}} = \frac{4}{9} \]

Final Answer