Which Graph Represents The Equation ( X − 1 2 ) 2 + ( Y + 5 2 ) 2 = 1 4 \left(x-\frac{1}{2}\right)^2+\left(y+\frac{5}{2}\right)^2=\frac{1}{4} ( X − 2 1 ​ ) 2 + ( Y + 2 5 ​ ) 2 = 4 1 ​ ?

The equation $\left(x-\frac{1}{2}\right)2+\left(y+\frac{5}{2}\right)2=\frac{1}{4}$ represents a circle in the Cartesian coordinate system. To identify the graph that represents this equation, we need to analyze its components and understand the properties of a circle.

The General Equation of a Circle

The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

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

By comparing this general equation with the given equation, we can identify the center and radius of the circle.

Identifying the Center and Radius

Comparing the given equation with the general equation, we can see that:

$h = \frac{1}{2}$ (the x-coordinate of the center) $k = -\frac{5}{2}$ (the y-coordinate of the center) $r^2 = \frac{1}{4}$ (the square of the radius)

Taking the square root of both sides of the equation, we get:

$r = \frac{1}{2}$ (the radius of the circle)

Graphing the Circle

Now that we have identified the center and radius of the circle, we can graph it. The center of the circle is at the point $\left(\frac{1}{2}, -\frac{5}{2}\right)$, and the radius is $\frac{1}{2}$.

To graph the circle, we can start by plotting the center point. Then, we can draw a circle with a radius of $\frac{1}{2}$ centered at the point $\left(\frac{1}{2}, -\frac{5}{2}\right)$.

Which Graph Represents the Equation?

Based on our analysis, the graph that represents the equation $\left(x-\frac{1}{2}\right)^2+\left(y+\frac{5}{2}\right)^2=\frac{1}{4}$ is a circle with center $\left(\frac{1}{2}, -\frac{5}{2}\right)$ and radius $\frac{1}{2}$.

Conclusion

In this discussion, we analyzed the equation $\left(x-\frac{1}{2}\right)^2+\left(y+\frac{5}{2}\right)^2=\frac{1}{4}$ and identified the center and radius of the circle it represents. We then graphed the circle and determined that the graph that represents the equation is a circle with center $\left(\frac{1}{2}, -\frac{5}{2}\right)$ and radius $\frac{1}{2}$.

Key Takeaways

• The equation $\left(x-\frac{1}{2}\right)^2+\left(y+\frac{5}{2}\right)^2=\frac{1}{4}$ represents a circle in the Cartesian coordinate system.
• The center of the circle is at the point $\left(\frac{1}{2}, -\frac{5}{2}\right)$.
• The radius of the circle is $\frac{1}{2}$.
• The graph that represents the equation is a circle with center $\left(\frac{1}{2}, -\frac{5}{2}\right)$ and radius $\frac{1}{2}$.

Further Reading

For more information on the equation of a circle and how to graph it, see the following resources:

• Wikipedia: Circle
• Math Open Reference: Circle
• Khan Academy: Circles

References

• [1] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
• [2] "Calculus" by Michael Spivak
• [3] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  Q&A: Understanding the Equation of a Circle =====================================================

In our previous discussion, we analyzed the equation $\left(x-\frac{1}{2}\right)^2+\left(y+\frac{5}{2}\right)^2=\frac{1}{4}$ and identified the center and radius of the circle it represents. In this Q&A article, we will answer some common questions related to the equation of a circle and provide additional insights.

Q: What is the general equation of a circle?

A: The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

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

Q: How do I identify the center and radius of a circle from its equation?

A: To identify the center and radius of a circle from its equation, you need to compare the equation with the general equation of a circle. The center of the circle is given by $(h, k)$, and the radius is given by $r = \sqrt{r^2}$.

Q: What is the significance of the center of a circle?

A: The center of a circle is the point around which the circle is symmetric. It is the point that is equidistant from all points on the circle.

Q: What is the significance of the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is a measure of the size of the circle.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center point and then draw a circle with a radius equal to the distance from the center point to any point on the circle.

Q: What are some common mistakes to avoid when graphing a circle?

A: Some common mistakes to avoid when graphing a circle include:

• Plotting the center point incorrectly
• Drawing the circle with the wrong radius
• Not using a compass or other drawing tool to ensure accuracy

Q: Can I use a calculator to graph a circle?

A: Yes, you can use a calculator to graph a circle. Many graphing calculators have a built-in function for graphing circles.

Q: What are some real-world applications of the equation of a circle?

A: The equation of a circle has many real-world applications, including:

• Designing circular shapes for architecture and engineering
• Modeling the motion of objects in physics and engineering
• Analyzing data in statistics and data analysis

Q: How do I use the equation of a circle to solve problems?

A: To use the equation of a circle to solve problems, you need to:

• Identify the center and radius of the circle
• Use the equation to find the distance between two points on the circle
• Use the equation to find the area of the circle
• Use the equation to find the circumference of the circle

Conclusion

In this Q&A article, we answered some common questions related to the equation of a circle and provided additional insights. We hope that this article has been helpful in understanding the equation of a circle and how to use it to solve problems.

Key Takeaways

• The general equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$.
• The center of a circle is given by $(h, k)$.
• The radius of a circle is given by $r = \sqrt{r^2}$.
• The equation of a circle can be used to graph a circle.
• The equation of a circle has many real-world applications.

Further Reading

For more information on the equation of a circle and how to use it to solve problems, see the following resources:

• Wikipedia: Circle
• Math Open Reference: Circle
• Khan Academy: Circles

References

• [1] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
• [2] "Calculus" by Michael Spivak
• [3] "Geometry: A Comprehensive Introduction" by Dan Pedoe