The equation of a circle is given below. Identify the center and the radius. Then graph the circle.=+x2+4y−y2210

To analyze the given equation of the circle, let's first rewrite it for clarity:

The equation provided is: $x^2 + 4y - y^2 = 210$

1. Break Down the Problem

We need to:

1. Rearrange the equation to standard form of a circle.
2. Identify the center and radius from the standard form.
3. Graph the circle.

2. Relevant Concepts

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.

3. Analysis and Detail

Starting with the given equation: $x^2 - y^2 + 4y = 210$ We need to rearrange and complete the square for the $y$ terms.

1. Rearranging the equation: $x^2 - (y^2 - 4y) = 210$

2. Completing the square on $y^2 - 4y$:

   • Take $-4/2 = -2$ and square it to get $4$.
   • Rewrite: $x^2 - (y^2 - 4y + 4 - 4) = 210$
   • This simplifies to: $x^2 - (y - 2)^2 + 4 = 210$
   • Further simplifying gives: $x^2 - (y - 2)^2 = 206$

3. Rearranging into standard form: $(y - 2)^2 + x^2 = 206$

Now, we can identify the center and radius:

• Center: $(0, 2)$
• Radius: $r = \sqrt{206}$

4. Verify and Summarize

To summarize:

• The center of the circle is at $(0, 2)$.
• The radius is $\sqrt{206}$.

Final Answer

• Center: $(0, 2)$
• Radius: $\sqrt{206}$

You can graph this circle centered at $(0, 2)$ with a radius of approximately $14.35$.