Which Of The Following Circles Lie Completely Within The Third Quadrant? Check All That Apply.A. { (x+5) 2+(y+0) 2=7$}$ B. { (x+7) 2+(y+7) 2=4$}$ C. { (x+3) 2+(y+9) 2=82$}$ D. { (x+12) 2+(y+9) 2=9$}$
In coordinate geometry, the third quadrant is defined by the region where both the x-coordinate and the y-coordinate are negative. This means that any point (x, y) in the third quadrant will have a negative x-value and a negative y-value. In this article, we will explore which of the given circles lie completely within the third quadrant.
What is the Third Quadrant?
The third quadrant is one of the four quadrants in the Cartesian coordinate system. It is located in the bottom-left corner of the coordinate plane and is defined by the region where both the x-coordinate and the y-coordinate are negative. The third quadrant is bounded by the x-axis and the y-axis, and it is the region where the x-coordinate is less than 0 and the y-coordinate is less than 0.
Analyzing the Given Circles
To determine which of the given circles lie completely within the third quadrant, we need to analyze each circle individually. We will examine the center and radius of each circle to determine if it lies within the third quadrant.
Circle A: {(x+5)2+(y+0)2=7$}$
The center of Circle A is at (-5, 0), which is on the x-axis. Since the y-coordinate is 0, the center of the circle is on the boundary between the third quadrant and the second quadrant. The radius of the circle is √7, which is approximately 2.65. Since the center of the circle is on the x-axis, the circle intersects the x-axis at two points, one in the second quadrant and one in the third quadrant. Therefore, Circle A does not lie completely within the third quadrant.
Circle B: {(x+7)2+(y+7)2=4$}$
The center of Circle B is at (-7, -7), which is in the third quadrant. The radius of the circle is √4, which is 2. Since the center of the circle is in the third quadrant and the radius is less than the distance from the center to the x-axis, the circle lies completely within the third quadrant.
Circle C: {(x+3)2+(y+9)2=82$}$
The center of Circle C is at (-3, -9), which is in the third quadrant. The radius of the circle is √82, which is approximately 9.05. Since the center of the circle is in the third quadrant and the radius is less than the distance from the center to the x-axis, the circle lies completely within the third quadrant.
Circle D: {(x+12)2+(y+9)2=9$}$
The center of Circle D is at (-12, -9), which is in the third quadrant. The radius of the circle is √9, which is 3. Since the center of the circle is in the third quadrant and the radius is less than the distance from the center to the x-axis, the circle lies completely within the third quadrant.
Conclusion
In conclusion, the circles that lie completely within the third quadrant are:
- Circle B: {(x+7)2+(y+7)2=4$}$
- Circle C: {(x+3)2+(y+9)2=82$}$
- Circle D: {(x+12)2+(y+9)2=9$}$
In the previous article, we explored which of the given circles lie completely within the third quadrant. In this article, we will answer some frequently asked questions (FAQs) about circles in the third quadrant.
Q: What is the significance of the third quadrant in coordinate geometry?
A: The third quadrant is one of the four quadrants in the Cartesian coordinate system. It is located in the bottom-left corner of the coordinate plane and is defined by the region where both the x-coordinate and the y-coordinate are negative. The third quadrant is bounded by the x-axis and the y-axis, and it is the region where the x-coordinate is less than 0 and the y-coordinate is less than 0.
Q: How do I determine if a circle lies completely within the third quadrant?
A: To determine if a circle lies completely within the third quadrant, you need to analyze the center and radius of the circle. If the center of the circle is in the third quadrant and the radius is less than the distance from the center to the x-axis, then the circle lies completely within the third quadrant.
Q: What is the relationship between the center of a circle and the third quadrant?
A: The center of a circle can be in any of the four quadrants, but if it is in the third quadrant, then the circle can lie completely within the third quadrant. The center of a circle is the point that is equidistant from all points on the circle.
Q: Can a circle have its center on the x-axis and still lie completely within the third quadrant?
A: No, a circle cannot have its center on the x-axis and still lie completely within the third quadrant. If the center of a circle is on the x-axis, then the circle intersects the x-axis at two points, one in the second quadrant and one in the third quadrant. Therefore, the circle does not lie completely within the third quadrant.
Q: Can a circle have its center in the third quadrant and still intersect the x-axis?
A: Yes, a circle can have its center in the third quadrant and still intersect the x-axis. If the radius of the circle is greater than the distance from the center to the x-axis, then the circle intersects the x-axis at two points, one in the second quadrant and one in the third quadrant.
Q: What is the relationship between the radius of a circle and the third quadrant?
A: The radius of a circle is the distance from the center of the circle to any point on the circle. If the radius of a circle is less than the distance from the center to the x-axis, then the circle lies completely within the third quadrant.
Q: Can a circle have a radius greater than the distance from the center to the x-axis and still lie completely within the third quadrant?
A: No, a circle cannot have a radius greater than the distance from the center to the x-axis and still lie completely within the third quadrant. If the radius of a circle is greater than the distance from the center to the x-axis, then the circle intersects the x-axis at two points, one in the second quadrant and one in the third quadrant.
Conclusion
In conclusion, the third quadrant is an important region in coordinate geometry, and understanding the relationship between the center and radius of a circle and the third quadrant is crucial in determining if a circle lies completely within the third quadrant. By analyzing the center and radius of a circle, you can determine if it lies completely within the third quadrant.
Additional Resources
Frequently Asked Questions (FAQs) About Circles in the Third Quadrant:
- Q: What is the significance of the third quadrant in coordinate geometry?
- A: The third quadrant is one of the four quadrants in the Cartesian coordinate system. It is located in the bottom-left corner of the coordinate plane and is defined by the region where both the x-coordinate and the y-coordinate are negative.
- Q: How do I determine if a circle lies completely within the third quadrant?
- A: To determine if a circle lies completely within the third quadrant, you need to analyze the center and radius of the circle. If the center of the circle is in the third quadrant and the radius is less than the distance from the center to the x-axis, then the circle lies completely within the third quadrant.
- Q: What is the relationship between the center of a circle and the third quadrant?
- A: The center of a circle can be in any of the four quadrants, but if it is in the third quadrant, then the circle can lie completely within the third quadrant.
- Q: Can a circle have its center on the x-axis and still lie completely within the third quadrant?
- A: No, a circle cannot have its center on the x-axis and still lie completely within the third quadrant.
- Q: Can a circle have its center in the third quadrant and still intersect the x-axis?
- A: Yes, a circle can have its center in the third quadrant and still intersect the x-axis.
- Q: What is the relationship between the radius of a circle and the third quadrant?
- A: The radius of a circle is the distance from the center of the circle to any point on the circle. If the radius of a circle is less than the distance from the center to the x-axis, then the circle lies completely within the third quadrant.
- Q: Can a circle have a radius greater than the distance from the center to the x-axis and still lie completely within the third quadrant?
- A: No, a circle cannot have a radius greater than the distance from the center to the x-axis and still lie completely within the third quadrant.