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

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

Transcript text: Determine the center and radius of the circle. Write your answers as si \[ \left(x+\frac{4}{3}\right)^{2}+\left(y-\frac{1}{5}\right)^{2}=\frac{16}{81} \]

Solution

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.

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

Step 1: Identify the Center

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: \[ \left( -\frac{4}{3}, \frac{1}{5} \right) \]

Step 2: Determine the Radius

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: \[ r = \sqrt{\frac{16}{81}} = \frac{4}{9} \]