Which Of The Following Circles Have Their Centers In The Second Quadrant? Check All That Apply.A. \[$(x+12)^2+(y-9)^2=19\$\]B. \[$(x-5)^2+(y+5)^2=9\$\]C. \[$(x-2)^2+(y+7)^2=64\$\]D. \[$(x+3)^2+(y-2)^2=8\$\]

In mathematics, particularly in geometry and algebra, circles are an essential concept. A circle is defined as the set of all points in a plane that are at a given distance from a given point, known as the center. The equation of a circle 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.

When dealing with circles, it's crucial to understand the location of their centers. In this article, we will focus on identifying circles with centers in the second quadrant. The second quadrant is the region of the coordinate plane where both $x$ and $y$ coordinates are negative.

Understanding Quadrants

Before we dive into the problem, let's quickly review the quadrants in the coordinate plane.

• First Quadrant: $x > 0$, $y > 0$
• Second Quadrant: $x < 0$, $y > 0$
• Third Quadrant: $x < 0$, $y < 0$
• Fourth Quadrant: $x > 0$, $y < 0$

Analyzing the Given Equations

Now, let's analyze the given equations to determine which circles have their centers in the second quadrant.

Equation A: $(x+12)^2+(y-9)^2=19$

To find the center of this circle, we need to look at the values inside the parentheses. The center is given by $(-12, 9)$. Since the $x$-coordinate is negative and the $y$-coordinate is positive, the center of this circle lies in the second quadrant.

Equation B: $(x-5)^2+(y+5)^2=9$

The center of this circle is given by $(5, -5)$. Since the $x$-coordinate is positive and the $y$-coordinate is negative, the center of this circle lies in the fourth quadrant.

Equation C: $(x-2)^2+(y+7)^2=64$

The center of this circle is given by $(2, -7)$. Since the $x$-coordinate is positive and the $y$-coordinate is negative, the center of this circle lies in the fourth quadrant.

Equation D: $(x+3)^2+(y-2)^2=8$

The center of this circle is given by $(-3, 2)$. Since the $x$-coordinate is negative and the $y$-coordinate is positive, the center of this circle lies in the second quadrant.

Conclusion

In conclusion, the circles with centers in the second quadrant are:

• Equation A: $(x+12)^2+(y-9)^2=19$
• Equation D: $(x+3)^2+(y-2)^2=8$

These two circles have their centers in the second quadrant, as their $x$-coordinates are negative and their $y$-coordinates are positive.

Key Takeaways

• The equation of a circle 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.
• The second quadrant is the region of the coordinate plane where both $x$ and $y$ coordinates are negative.
• To identify circles with centers in the second quadrant, look for circles with negative $x$-coordinates and positive $y$-coordinates.

In the previous article, we discussed how to identify circles with centers in the second quadrant. However, we understand that you may still have some questions or concerns. In this article, we will address some of the most frequently asked questions about circles with centers in the second quadrant.

Q: What is the second quadrant in the coordinate plane?

A: The second quadrant is the region of the coordinate plane where both $x$ and $y$ coordinates are negative. It is one of the four quadrants in the coordinate plane, and it is located in the upper left corner.

Q: How do I identify a circle with its center in the second quadrant?

A: To identify a circle with its center in the second quadrant, look for a circle with a negative $x$-coordinate and a positive $y$-coordinate. This means that the center of the circle will be located in the upper left corner of the coordinate plane.

Q: What is the equation of a circle with its center in the second quadrant?

A: The equation of a circle with its center in the second quadrant 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. The center of the circle will have a negative $x$-coordinate and a positive $y$-coordinate.

Q: Can a circle have its center in the second quadrant and still have a positive radius?

A: Yes, a circle can have its center in the second quadrant and still have a positive radius. The radius of a circle is the distance from the center of the circle to any point on the circle. As long as the radius is positive, the circle will still be a valid circle.

Q: How do I graph a circle with its center in the second quadrant?

A: To graph a circle with its center in the second quadrant, start by plotting the center of the circle. Since the center is in the second quadrant, the $x$-coordinate will be negative and the $y$-coordinate will be positive. Then, use the radius to draw the circle. Make sure to include all points that are a distance of $r$ units from the center of the circle.

Q: Can a circle with its center in the second quadrant intersect with other circles?

A: Yes, a circle with its center in the second quadrant can intersect with other circles. In fact, circles can intersect with each other in many different ways, including tangent circles, intersecting circles, and concentric circles.

Q: How do I find the intersection points of two circles with their centers in the second quadrant?

A: To find the intersection points of two circles with their centers in the second quadrant, you can use the equation of a circle and the distance formula. The distance formula is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the centers of the two circles.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions about circles with centers in the second quadrant. We hope that this article has provided you with a better understanding of how to identify and graph circles with their centers in the second quadrant.

Key Takeaways

• The second quadrant is the region of the coordinate plane where both $x$ and $y$ coordinates are negative.
• To identify a circle with its center in the second quadrant, look for a circle with a negative $x$-coordinate and a positive $y$-coordinate.
• The equation of a circle with its center in the second quadrant 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.
• A circle can have its center in the second quadrant and still have a positive radius.
• To graph a circle with its center in the second quadrant, start by plotting the center of the circle and then use the radius to draw the circle.

By understanding the basics of circles with centers in the second quadrant, you can apply this knowledge to a wide range of mathematical problems and real-world applications.