Complete The Square And Write The Given Equation In Standard Form. Then Give The Center And Radius Of The Circle And Graph The Equation.$ X^2 + Y^2 + 8x - 2y - 19 = 0 }$- Equation In Standard Form { \quad$ $ - Center:

Introduction

In mathematics, a circle is a set of points that are equidistant from a central point called the center. The equation of a circle in standard form is given by ${(x - h)^2 + (y - k)^2 = r^2}$, where ${(h, k)}$ is the center of the circle and ${r}$ is the radius. In this article, we will complete the square and write the given equation in standard form. We will also find the center and radius of the circle and graph the equation.

Completing the Square

To complete the square, we need to rewrite the given equation in a perfect square form. The given equation is ${x^2 + y^2 + 8x - 2y - 19 = 0}$. We can start by grouping the ${x}$ terms and ${y}$ terms separately.

Grouping the Terms

${x^2 + 8x + y^2 - 2y = 19}$

Completing the Square for x

To complete the square for ${x}$, we need to add ${\left(\frac{8}{2}\right)^2 = 16}$ to both sides of the equation.

${x^2 + 8x + 16 + y^2 - 2y = 19 + 16}$

Completing the Square for y

To complete the square for ${y}$, we need to add ${\left(\frac{-2}{2}\right)^2 = 1}$ to both sides of the equation.

${x^2 + 8x + 16 + y^2 - 2y + 1 = 19 + 16 + 1}$

Simplifying the Equation

Now, we can simplify the equation by combining like terms.

${(x + 4)^2 + (y - 1)^2 = 36}$

Writing the Equation in Standard Form

The equation is now in standard form, which is ${(x - h)^2 + (y - k)^2 = r^2}$. Comparing this with our equation, we can see that the center of the circle is ${(-4, 1)}$ and the radius is ${\sqrt{36} = 6}$.

Center and Radius

The center of the circle is ${(-4, 1)}$ and the radius is ${6}$.

Graphing the Equation

To graph the equation, we can use the center and radius to draw a circle. The center of the circle is ${(-4, 1)}$, so we can start by drawing a point at this location. The radius of the circle is ${6}$, so we can draw a circle with a radius of ${6}$ centered at ${(-4, 1)}$.

Graphing the Circle

Here is a graph of the circle:

Graph of the Circle

The graph of the circle is a circle with a center at ${(-4, 1)}$ and a radius of ${6}$.

Conclusion

In this article, we completed the square and wrote the given equation in standard form. We found the center and radius of the circle and graphed the equation. The center of the circle is ${(-4, 1)}$ and the radius is ${6}$. The graph of the circle is a circle with a center at ${(-4, 1)}$ and a radius of ${6}$.

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Graphing a Circle" by Math Is Fun

Discussion

Introduction

In our previous article, we completed the square and wrote the given equation in standard form. We found the center and radius of the circle and graphed the equation. In this article, we will answer some frequently asked questions about completing the square and graphing a circle equation.

Q&A

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a perfect square form. It involves adding and subtracting a constant term to create a perfect square trinomial.

Q: Why do we need to complete the square?

A: We need to complete the square to write the equation of a circle in standard form. The standard form of a circle equation is ${(x - h)^2 + (y - k)^2 = r^2}$, where ${(h, k)}$ is the center of the circle and ${r}$ is the radius.

Q: How do we complete the square?

A: To complete the square, we need to group the ${x}$ terms and ${y}$ terms separately. Then, we need to add and subtract a constant term to create a perfect square trinomial.

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

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

Q: How do we find the center and radius of a circle?

A: To find the center and radius of a circle, we need to complete the square and write the equation in standard form. The center of the circle is ${(h, k)}$ and the radius is ${r}$.

Q: How do we graph a circle?

A: To graph a circle, we need to draw a point at the center of the circle and then draw a circle with a radius of ${r}$ centered at the point.

Q: What is the difference between a circle and an ellipse?

A: A circle is a set of points that are equidistant from a central point, while an ellipse is a set of points that are equidistant from two central points.

Q: How do we find the equation of an ellipse?

A: To find the equation of an ellipse, we need to use the formula ${\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1}$, where ${(h, k)}$ is the center of the ellipse and ${a}$ and ${b}$ are the lengths of the semi-major and semi-minor axes.

Q: Can we graph an ellipse?

A: Yes, we can graph an ellipse by drawing a point at the center of the ellipse and then drawing an ellipse with a semi-major axis of ${a}$ and a semi-minor axis of ${b}$ centered at the point.

Conclusion

In this article, we answered some frequently asked questions about completing the square and graphing a circle equation. We hope that this article has been helpful in understanding the concepts of completing the square and graphing a circle equation.

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Graphing a Circle" by Math Is Fun
• [3] "Equation of an Ellipse" by Mathway

Discussion

What is your favorite mathematical concept? How do you use completing the square in your math problems? What is the most challenging part of graphing a circle?