Find An Equation Of The Circle That Has Center { (-1,5)$}$ And Passes Through { (-5,1)$}$.

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point known as the center. The distance between the center and any point on the circle is called the radius. Given the center of a circle and a point that lies on the circle, we can find the equation of the circle using the distance formula. In this article, we will discuss how to find the equation of a circle with a given center and a point that passes through it.

The General Equation of a Circle

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

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

where ${(x, y)}$ are the coordinates of any point on the circle.

Finding the Equation of the Circle

We are given that the center of the circle is ${(-1, 5)}$ and the point ${(-5, 1)}$ lies on the circle. We can use the distance formula to find the radius of the circle.

The Distance Formula

The distance formula is given by:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

where ${d}$ is the distance between the points ${(x_1, y_1)}$ and ${(x_2, y_2)}$.

Finding the Radius of the Circle

We can use the distance formula to find the radius of the circle by substituting the coordinates of the center and the point ${(-5, 1)}$ into the formula.

${r = \sqrt{(-5 - (-1))^2 + (1 - 5)^2}}$ ${r = \sqrt{(-4)^2 + (-4)^2}}$ ${r = \sqrt{16 + 16}}$ ${r = \sqrt{32}}$ ${r = 4\sqrt{2}}$

Substituting the Values into the General Equation

Now that we have the radius of the circle, we can substitute the values of the center and the radius into the general equation of the circle.

${(x - (-1))^2 + (y - 5)^2 = (4\sqrt{2})^2}$ ${(x + 1)^2 + (y - 5)^2 = 32}$

Simplifying the Equation

We can simplify the equation by expanding the squared terms.

${(x + 1)^2 = x^2 + 2x + 1}$ ${(y - 5)^2 = y^2 - 10y + 25}$

Substituting the Simplified Terms into the Equation

We can substitute the simplified terms into the equation.

${x^2 + 2x + 1 + y^2 - 10y + 25 = 32}$

Combining Like Terms

We can combine like terms in the equation.

${x^2 + y^2 + 2x - 10y + 26 = 32}$

Subtracting 32 from Both Sides

We can subtract 32 from both sides of the equation to isolate the terms on the left-hand side.

${x^2 + y^2 + 2x - 10y - 6 = 0}$

Conclusion

In this article, we discussed how to find the equation of a circle with a given center and a point that passes through it. We used the distance formula to find the radius of the circle and then substituted the values of the center and the radius into the general equation of the circle. We simplified the equation and combined like terms to obtain the final equation of the circle.

Final Equation of the Circle

The final equation of the circle is:

${x^2 + y^2 + 2x - 10y - 6 = 0}$

This equation represents a circle with center ${(-1, 5)}$ and radius ${4\sqrt{2}}$.

Introduction

In the previous article, we discussed how to find the equation of a circle with a given center and a point that passes through it. In this article, we will answer some frequently asked questions about finding the equation of a circle.

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 find the radius of the circle?

A: You can use the distance formula to find the radius of the circle. The distance formula is given by:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

where ${d}$ is the distance between the points ${(x_1, y_1)}$ and ${(x_2, y_2)}$.

Q: What if the point is not on the circle?

A: If the point is not on the circle, you will not be able to find the radius of the circle using the distance formula. In this case, you will need to use a different method to find the equation of the circle.

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

A: Yes, you can use the equation of the circle to find the center and radius. To do this, you will need to complete the square on the equation.

Q: How do I complete the square on the equation?

A: To complete the square on the equation, you will need to add and subtract the square of half the coefficient of the x-term and the y-term.

Q: What is the final equation of the circle?

A: The final equation of the circle is:

${x^2 + y^2 + 2x - 10y - 6 = 0}$

This equation represents a circle with center ${(-1, 5)}$ and radius ${4\sqrt{2}}$.

Q: Can I use the equation of the circle to find the distance between two points on the circle?

A: Yes, you can use the equation of the circle to find the distance between two points on the circle. To do this, you will need to use the distance formula.

Q: How do I use the equation of the circle to find the distance between two points on the circle?

A: To use the equation of the circle to find the distance between two points on the circle, you will need to substitute the coordinates of the two points into the equation and solve for the distance.

Q: What if the equation of the circle is not in the standard form?

A: If the equation of the circle is not in the standard form, you will need to complete the square on the equation to put it in the standard form.

Q: Can I use the equation of the circle to find the area of the circle?

A: Yes, you can use the equation of the circle to find the area of the circle. To do this, you will need to use the formula for the area of a circle.

Q: How do I use the equation of the circle to find the area of the circle?

A: To use the equation of the circle to find the area of the circle, you will need to substitute the radius of the circle into the formula for the area of a circle.

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is:

${A = \pi r^2}$

where ${A}$ is the area of the circle and ${r}$ is the radius of the circle.

Conclusion

In this article, we answered some frequently asked questions about finding the equation of a circle. We discussed how to find the radius of the circle, how to complete the square on the equation, and how to use the equation of the circle to find the distance between two points on the circle and the area of the circle.