Which Of The Following Circles Lie Completely Within The Fourth Quadrant? Check All That Apply.A. { (x-9) 2+(y+9) 2=16$}$B. { (x-12) 2+(y+0) 2=72$}$C. { (x-5) 2+(y+5) 2=9$}$D. { (x-2) 2+(y+7) 2=64$}$
In the coordinate plane, the fourth quadrant is defined as the region where both x and y coordinates are negative. To determine which of the given circles lie completely within the fourth quadrant, we need to analyze the equations of each circle and identify their positions in relation to the x and y axes.
Understanding Circle Equations
A circle with center (h, k) and radius r has the equation (x - h)^2 + (y - k)^2 = r^2. By examining the given equations, we can identify the center and radius of each circle.
A. (x-9)^2 + (y+9)^2 = 16
- Center: (9, -9)
- Radius: 4
B. (x-12)^2 + (y+0)^2 = 72
- Center: (12, 0)
- Radius: √72 = √(36*2) = 6√2
C. (x-5)^2 + (y+5)^2 = 9
- Center: (5, -5)
- Radius: 3
D. (x-2)^2 + (y+7)^2 = 64
- Center: (2, -7)
- Radius: √64 = √(16*4) = 8
Analyzing Circle Positions
To determine which circles lie completely within the fourth quadrant, we need to examine the x and y coordinates of their centers.
- A circle lies in the fourth quadrant if its center has both x and y coordinates negative.
- A circle lies on the x-axis if its center has a y-coordinate of 0.
- A circle lies on the y-axis if its center has an x-coordinate of 0.
Circle A
The center of Circle A is at (9, -9), which means it lies in the fourth quadrant.
Circle B
The center of Circle B is at (12, 0), which means it lies on the x-axis. Since the x-coordinate is positive, Circle B does not lie in the fourth quadrant.
Circle C
The center of Circle C is at (5, -5), which means it lies in the fourth quadrant.
Circle D
The center of Circle D is at (2, -7), which means it lies in the fourth quadrant.
Conclusion
Based on the analysis of the circle equations and their positions in relation to the x and y axes, we can conclude that the following circles lie completely within the fourth quadrant:
- Circle A: (x-9)^2 + (y+9)^2 = 16
- Circle C: (x-5)^2 + (y+5)^2 = 9
- Circle D: (x-2)^2 + (y+7)^2 = 64
In the previous article, we discussed how to identify circles that lie completely within the fourth quadrant. Here are some frequently asked questions (FAQs) about circles in the fourth quadrant, along with their answers.
Q: What is the fourth quadrant in the coordinate plane?
A: The fourth 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, with the other three quadrants being the first quadrant (x > 0, y > 0), the second quadrant (x < 0, y > 0), and the third quadrant (x < 0, y < 0).
Q: How do I determine if a circle lies in the fourth quadrant?
A: To determine if a circle lies in the fourth quadrant, you need to examine the x and y coordinates of its center. If both x and y coordinates are negative, then the circle lies in the fourth quadrant.
Q: What is the equation of a circle in the fourth quadrant?
A: The equation of a circle in the fourth quadrant is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius.
Q: Can a circle have its center on the x-axis or y-axis?
A: Yes, a circle can have its center on the x-axis or y-axis. If the center has a y-coordinate of 0, then the circle lies on the x-axis. If the center has an x-coordinate of 0, then the circle lies on the y-axis.
Q: How do I determine if a circle lies on the x-axis or y-axis?
A: To determine if a circle lies on the x-axis or y-axis, you need to examine the x and y coordinates of its center. If the y-coordinate is 0, then the circle lies on the x-axis. If the x-coordinate is 0, then the circle lies on the y-axis.
Q: Can a circle have its center in the fourth quadrant and also lie on the x-axis or y-axis?
A: No, a circle cannot have its center in the fourth quadrant and also lie on the x-axis or y-axis. If a circle lies on the x-axis, then its center must have a y-coordinate of 0, which means it cannot have both x and y coordinates negative. Similarly, if a circle lies on the y-axis, then its center must have an x-coordinate of 0, which means it cannot have both x and y coordinates negative.
Q: How do I find the radius of a circle in the fourth quadrant?
A: To find the radius of a circle in the fourth quadrant, you need to examine the equation of the circle. The radius is given by the square root of the constant term on the right-hand side of the equation.
Q: Can a circle have a negative radius?
A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value, representing the distance from the center to the edge of the circle.
Q: How do I graph a circle in the fourth quadrant?
A: To graph a circle in the fourth quadrant, you need to plot the center of the circle and then draw a circle with the given radius. Make sure to use a compass or a circle template to draw the circle accurately.
Conclusion
In this article, we have answered some frequently asked questions about circles in the fourth quadrant. We have discussed how to determine if a circle lies in the fourth quadrant, how to find the radius of a circle, and how to graph a circle in the fourth quadrant. We hope that this article has been helpful in clarifying any doubts you may have had about circles in the fourth quadrant.