Find The Center And Radius Of The Circle With This Equation:$x^2 + 16x + 64 + Y^2 - 2y + 1 = 144$Center: $(\ \square\ , \ \square\ $\]Radius: $\square$ Units

Understanding the Circle Equation

The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ is the radius. However, the given equation is not in standard form, and we need to manipulate it to find the center and radius.

Manipulating the Given Equation

The given equation is $x^2 + 16x + 64 + y^2 - 2y + 1 = 144$. To convert it to standard form, we need to complete the square for both the $x$ and $y$ terms.

Completing the Square for the $x$ Terms

We start by grouping the $x$ terms: $x^2 + 16x$. To complete the square, we add $(16/2)^2 = 8^2 = 64$ to both sides of the equation.

x^2 + 16x + 64 + y^2 - 2y + 1 = 144
(x^2 + 16x + 64) + y^2 - 2y + 1 = 144 + 64
(x + 8)^2 + y^2 - 2y + 1 = 208

Completing the Square for the $y$ Terms

Next, we group the $y$ terms: $y^2 - 2y$. To complete the square, we add $(-2/2)^2 = 1^2 = 1$ to both sides of the equation.

(x + 8)^2 + y^2 - 2y + 1 = 208
(x + 8)^2 + (y^2 - 2y + 1) = 208 + 1
(x + 8)^2 + (y - 1)^2 = 209

Finding the Center and Radius

Now that we have the equation in standard form, we can easily identify the center and radius.

Center

The center of the circle is given by $(h, k)$, where $h$ is the value inside the parentheses of the $x$ term, and $k$ is the value inside the parentheses of the $y$ term. In this case, $h = -8$ and $k = 1$, so the center of the circle is $(-8, 1)$.

Radius

The radius of the circle is given by $r = \sqrt{r^2}$, where $r^2$ is the value on the right-hand side of the equation. In this case, $r^2 = 209$, so the radius of the circle is $\sqrt{209}$ units.

Conclusion

In this article, we have shown how to find the center and radius of a circle given its equation. We started by manipulating the given equation to convert it to standard form, and then we completed the square for both the $x$ and $y$ terms. Finally, we identified the center and radius of the circle from the standard form equation.

Example Problems

Problem 1

Find the center and radius of the circle with the equation $x^2 + 12x + 36 + y^2 - 4y + 4 = 100$.

Solution

To find the center and radius, we need to complete the square for both the $x$ and $y$ terms.

x^2 + 12x + 36 + y^2 - 4y + 4 = 100
(x^2 + 12x + 36) + y^2 - 4y + 4 = 100 + 36
(x + 6)^2 + y^2 - 4y + 4 = 136
(x + 6)^2 + (y^2 - 4y + 4) = 136 + 4
(x + 6)^2 + (y - 2)^2 = 140

The center of the circle is $(h, k) = (-6, 2)$, and the radius is $r = \sqrt{140}$ units.

Problem 2

Find the center and radius of the circle with the equation $x^2 - 10x + 25 + y^2 + 6y + 9 = 49$.

Solution

To find the center and radius, we need to complete the square for both the $x$ and $y$ terms.

x^2 - 10x + 25 + y^2 + 6y + 9 = 49
(x^2 - 10x + 25) + y^2 + 6y + 9 = 49 + 25
(x - 5)^2 + y^2 + 6y + 9 = 74
(x - 5)^2 + (y^2 + 6y + 9) = 74 + 9
(x - 5)^2 + (y + 3)^2 = 83

The center of the circle is $(h, k) = (5, -3)$, and the radius is $r = \sqrt{83}$ units.

Final Thoughts

In this article, we have shown how to find the center and radius of a circle given its equation. We have also provided example problems to help illustrate the process. By following these steps, you can easily find the center and radius of a circle, even if the equation is not in standard form.

Q: What is the standard form of a circle equation?

A: The standard form of a circle equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ is the radius.

Q: How do I complete the square for the $x$ terms?

A: To complete the square for the $x$ terms, you need to add $(b/2)^2$ to both sides of the equation, where $b$ is the coefficient of the $x$ term. For example, if the equation is $x^2 + 6x + 1 = 0$, you would add $(6/2)^2 = 9$ to both sides.

Q: How do I complete the square for the $y$ terms?

A: To complete the square for the $y$ terms, you need to add $(c/2)^2$ to both sides of the equation, where $c$ is the coefficient of the $y$ term. For example, if the equation is $y^2 - 4y + 1 = 0$, you would add $(-4/2)^2 = 4$ to both sides.

Q: What is the center of the circle?

A: The center of the circle is given by $(h, k)$, where $h$ is the value inside the parentheses of the $x$ term, and $k$ is the value inside the parentheses of the $y$ term.

Q: How do I find the radius of the circle?

A: To find the radius of the circle, you need to take the square root of the value on the right-hand side of the equation. For example, if the equation is $(x - 3)^2 + (y - 4)^2 = 16$, the radius is $\sqrt{16} = 4$ units.

Q: Can I find the center and radius of a circle if the equation is not in standard form?

A: Yes, you can find the center and radius of a circle even if the equation is not in standard form. You just need to complete the square for both the $x$ and $y$ terms to convert the equation to standard form.

Q: What if I have a circle equation with a negative radius?

A: If you have a circle equation with a negative radius, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a calculator to find the center and radius of a circle?

A: Yes, you can use a calculator to find the center and radius of a circle. However, it's always a good idea to double-check your work by completing the square manually.

Q: What if I have a circle equation with a complex center?

A: If you have a circle equation with a complex center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I find the center and radius of a circle if the equation is a quadratic equation?

A: Yes, you can find the center and radius of a circle even if the equation is a quadratic equation. You just need to complete the square for both the $x$ and $y$ terms to convert the equation to standard form.

Q: What if I have a circle equation with a negative constant term?

A: If you have a circle equation with a negative constant term, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a graphing calculator to find the center and radius of a circle?

A: Yes, you can use a graphing calculator to find the center and radius of a circle. However, it's always a good idea to double-check your work by completing the square manually.

Q: What if I have a circle equation with a complex radius?

A: If you have a circle equation with a complex radius, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I find the center and radius of a circle if the equation is a parametric equation?

A: Yes, you can find the center and radius of a circle even if the equation is a parametric equation. You just need to convert the parametric equation to a standard form equation and then complete the square.

Q: What if I have a circle equation with a negative center?

A: If you have a circle equation with a negative center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a computer algebra system to find the center and radius of a circle?

A: Yes, you can use a computer algebra system to find the center and radius of a circle. However, it's always a good idea to double-check your work by completing the square manually.

Q: What if I have a circle equation with a complex center and radius?

A: If you have a circle equation with a complex center and radius, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I find the center and radius of a circle if the equation is a polar equation?

A: Yes, you can find the center and radius of a circle even if the equation is a polar equation. You just need to convert the polar equation to a standard form equation and then complete the square.

Q: What if I have a circle equation with a negative radius and center?

A: If you have a circle equation with a negative radius and center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a graphing calculator to find the center and radius of a circle with a complex center and radius?

A: No, you cannot use a graphing calculator to find the center and radius of a circle with a complex center and radius. In this case, you need to use a computer algebra system or complete the square manually.

Q: What if I have a circle equation with a negative constant term and a complex center and radius?

A: If you have a circle equation with a negative constant term and a complex center and radius, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I find the center and radius of a circle if the equation is a parametric equation with a complex center and radius?

A: Yes, you can find the center and radius of a circle even if the equation is a parametric equation with a complex center and radius. You just need to convert the parametric equation to a standard form equation and then complete the square.

Q: What if I have a circle equation with a negative center and a complex radius?

A: If you have a circle equation with a negative center and a complex radius, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a computer algebra system to find the center and radius of a circle with a complex center and radius?

A: Yes, you can use a computer algebra system to find the center and radius of a circle with a complex center and radius. However, it's always a good idea to double-check your work by completing the square manually.

Q: What if I have a circle equation with a negative constant term and a complex center?

A: If you have a circle equation with a negative constant term and a complex center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I find the center and radius of a circle if the equation is a polar equation with a complex center and radius?

A: Yes, you can find the center and radius of a circle even if the equation is a polar equation with a complex center and radius. You just need to convert the polar equation to a standard form equation and then complete the square.

Q: What if I have a circle equation with a negative radius and a complex center?

A: If you have a circle equation with a negative radius and a complex center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors.

Q: Can I use a graphing calculator to find the center and radius of a circle with a negative radius and a complex center?

A: No, you cannot use a graphing calculator to find the center and radius of a circle with a negative radius and a complex center. In this case, you need to use a computer algebra system or complete the square manually.

Q: What if I have a circle equation with a negative constant term and a negative radius and a complex center?

A: If you have a circle equation with a negative constant term and a negative radius and a complex center, it means that the circle is not a real circle. In this case, you need to re-examine the equation and check for any errors