Understanding the Original Circle

Oct 13, 2025 by windailys.com 2 views

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle in the 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 be dealing with a circle given by the equation $(x-3)^2+(y+1)^2=\frac{1}{4}$ . We will shift this circle left by 5 units and triple its radius to find the new center and radius.

Understanding the Original Circle

The original circle is given by the equation $(x-3)^2+(y+1)^2=\frac{1}{4}$ . To understand this circle, let's break down the equation. The equation is in the standard form of a circle, which is $(x-h)^2+(y-k)^2=r^2$ . Comparing this with the given equation, we can see that the center of the circle is at $(h,k)=(3,-1)$ and the radius is $r=\sqrt{\frac{1}{4}}=\frac{1}{2}$ .

Shifting the Circle Left by 5 Units

To shift the circle left by 5 units, we need to change the x-coordinate of the center of the circle. The new x-coordinate will be $3-5=-2$ . The y-coordinate of the center remains the same, which is $-1$ . Therefore, the new center of the circle is at $(-2,-1)$ .

Tripling the Radius

To triple the radius of the circle, we need to multiply the current radius by 3. The current radius is $\frac{1}{2}$ , so the new radius will be $3\times\frac{1}{2}=\frac{3}{2}$ .

Finding the New Equation of the Circle

Now that we have the new center and radius, we can find the new equation of the circle. The equation of a circle in the standard form is given by $(x-h)^2+(y-k)^2=r^2$ . Substituting the new center $(-2,-1)$ and the new radius $\frac{3}{2}$ , we get the new equation of the circle as $(x+2)^2+(y+1)^2=\left(\frac{3}{2}\right)^2$ .

Simplifying the New Equation

To simplify the new equation, we can expand the squared terms and combine like terms. Expanding the squared terms, we get $(x+2)^2=x^2+4x+4$ and $(y+1)^2=y^2+2y+1$ . Substituting these into the new equation, we get $x^2+4x+4+y^2+2y+1=\left(\frac{3}{2}\right)^2$ . Combining like terms, we get $x^2+y^2+4x+2y+5=\frac{9}{4}$ .

Conclusion

In this article, we started with a circle given by the equation $(x-3)^2+(y+1)^2=\frac{1}{4}$ . We shifted this circle left by 5 units and tripled its radius to find the new center and radius. The new center of the circle is at $(-2,-1)$ and the new radius is $\frac{3}{2}$ . We also found the new equation of the circle as $(x+2)^2+(y+1)^2=\left(\frac{3}{2}\right)^2$ and simplified it to $x^2+y^2+4x+2y+5=\frac{9}{4}$ .

Final Answer

The final answer is:

Center: $(-2,-1)$
Radius: $\frac{3}{2}$

Introduction

In our previous article, we discussed how to shift a circle left by 5 units and triple its radius. We found the new center and radius of the circle and simplified the new equation of the circle. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the center of the original circle?

A: The center of the original circle is at $(h,k)=(3,-1)$ .

Q: What is the radius of the original circle?

A: The radius of the original circle is $r=\sqrt{\frac{1}{4}}=\frac{1}{2}$ .

Q: How do I shift a circle left by 5 units?

A: To shift a circle left by 5 units, you need to change the x-coordinate of the center of the circle by subtracting 5 from it. The new x-coordinate will be $h-5$ .

Q: How do I triple the radius of a circle?

A: To triple the radius of a circle, you need to multiply the current radius by 3. The new radius will be $3r$ .

Q: What is the new equation of the circle after shifting it left by 5 units and tripling its radius?

A: The new equation of the circle is $(x+2)^2+(y+1)^2=\left(\frac{3}{2}\right)^2$ .

Q: How do I simplify the new equation of the circle?

A: To simplify the new equation of the circle, you need to expand the squared terms and combine like terms. Expanding the squared terms, you get $(x+2)^2=x^2+4x+4$ and $(y+1)^2=y^2+2y+1$ . Substituting these into the new equation, you get $x^2+4x+4+y^2+2y+1=\left(\frac{3}{2}\right)^2$ . Combining like terms, you get $x^2+y^2+4x+2y+5=\frac{9}{4}$ .

Q: What is the final answer for the center and radius of the new circle?

A: The final answer for the center and radius of the new circle is:

Center: $(-2,-1)$
Radius: $\frac{3}{2}$

Q: What is the significance of the new equation of the circle?

A: The new equation of the circle represents the new circle after shifting it left by 5 units and tripling its radius. It can be used to find the points on the new circle.

Q: How do I use the new equation of the circle to find the points on the new circle?

A: To use the new equation of the circle to find the points on the new circle, you need to substitute the x and y values into the equation and solve for the other variable. For example, if you want to find the points on the new circle with x-coordinate 0, you can substitute x=0 into the equation and solve for y.

Conclusion

In this article, we answered some frequently asked questions related to shifting a circle left by 5 units and tripling its radius. We provided the final answer for the center and radius of the new circle and explained the significance of the new equation of the circle. We also provided a step-by-step guide on how to use the new equation of the circle to find the points on the new circle.

Final Answer

The final answer is:

Center: $(-2,-1)$
Radius: $\frac{3}{2}$