The given equation is:

$\left( x - \frac{5}{2} \right)^2 + (y - 4)^2 = 2$

This is the equation of a circle in standard form:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius of the circle.

Identifying the parts:

1. Center: From the equation, $(h, k)$ represents the center. Here:

   • $h = \frac{5}{2} = 2.5$
   • $k = 4$

   Therefore, the center of the circle is $(2.5, 4)$.

2. Radius: The right-hand side of the equation is $r^2$. Here:

   • $r^2 = 2$
   • Therefore, $r = \sqrt{2}$ (approximately $1.414$).

3. Graph of the circle:

   • The center is located at $(2.5, 4)$.
   • The radius is $\sqrt{2}$, so the distance from the center to any point on the circle is approximately $1.414$ units.

Would you like me to graph this circle for you? Also, feel free to ask any further questions.

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Here are 5 related questions for you to explore:

1. How do you find the equation of a circle given three points on its circumference?
2. What is the significance of completing the square in deriving a circle equation?
3. How do you determine the tangent line to a circle at a given point?
4. How do you find the area of a sector of a circle given the radius and angle?
5. How would you write the equation of a circle if its center is at the origin?

Tip: When solving equations involving circles, knowing how to complete the square can be essential for converting general forms into standard forms.