Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which circles lie entirely within the fourth quadrant, let's examine the given equations of the circles in standard form:
1. Circle A: [tex]\((x-5)^2 + (y+7)^2 = 16\)[/tex]
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(\sqrt{16} = 4\)[/tex]
2. Circle B: [tex]\((x+3)^2 + (y-2)^2 = 25\)[/tex]
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(\sqrt{25} = 5\)[/tex]
3. Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(\sqrt{1} = 1\)[/tex]
4. Circle D: [tex]\((x-4)^2 + (y+2)^2 = 32\)[/tex]
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
The fourth quadrant is defined where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]. Also, for a circle to lie completely within the fourth quadrant, its top part should not cross beyond the x-axis (i.e., [tex]\(y + \text{radius} < 0\)[/tex]) and its left part should not cross beyond the y-axis (i.e., [tex]\(x - \text{radius} > 0\)[/tex]).
Now, we evaluate each circle:
### Circle A:
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(4\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(5 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-7 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y\ + \text{radius} = -7 + 4 = -3 > 0\)[/tex] (fails the [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle A does not lie completely within the fourth quadrant.
### Circle B:
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(5\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(-3 < 0\)[/tex] (fails [tex]\(x > 0\)[/tex] check)
- [tex]\(y\)[/tex]-coordinate: [tex]\(2 > 0\)[/tex] (fails [tex]\(y < 0\)[/tex] check)
Circle B does not lie completely within the fourth quadrant.
### Circle C:
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(1\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(3 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-4 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -4 + 1 = -3 < 0\)[/tex] (satisfies [tex]\(y + \text{radius} < 0\)[/tex] check)
- [tex]\(x - \text{radius} = 3 - 1 = 2 > 0\)[/tex] (satisfies [tex]\(x - \text{radius} > 0\)[/tex] check)
Circle C lies completely within the fourth quadrant.
### Circle D:
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(4 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-2 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -2 + 5.66 > 0\)[/tex] (fails [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle D does not lie completely within the fourth quadrant.
Summary:
The circle that lies completely within the fourth quadrant is:
- Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
Therefore, the correct answer is only Circle C, hence we submit "C."
1. Circle A: [tex]\((x-5)^2 + (y+7)^2 = 16\)[/tex]
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(\sqrt{16} = 4\)[/tex]
2. Circle B: [tex]\((x+3)^2 + (y-2)^2 = 25\)[/tex]
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(\sqrt{25} = 5\)[/tex]
3. Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(\sqrt{1} = 1\)[/tex]
4. Circle D: [tex]\((x-4)^2 + (y+2)^2 = 32\)[/tex]
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
The fourth quadrant is defined where [tex]\(x > 0\)[/tex] and [tex]\(y < 0\)[/tex]. Also, for a circle to lie completely within the fourth quadrant, its top part should not cross beyond the x-axis (i.e., [tex]\(y + \text{radius} < 0\)[/tex]) and its left part should not cross beyond the y-axis (i.e., [tex]\(x - \text{radius} > 0\)[/tex]).
Now, we evaluate each circle:
### Circle A:
- Center: [tex]\((5, -7)\)[/tex]
- Radius: [tex]\(4\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(5 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-7 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y\ + \text{radius} = -7 + 4 = -3 > 0\)[/tex] (fails the [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle A does not lie completely within the fourth quadrant.
### Circle B:
- Center: [tex]\((-3, 2)\)[/tex]
- Radius: [tex]\(5\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(-3 < 0\)[/tex] (fails [tex]\(x > 0\)[/tex] check)
- [tex]\(y\)[/tex]-coordinate: [tex]\(2 > 0\)[/tex] (fails [tex]\(y < 0\)[/tex] check)
Circle B does not lie completely within the fourth quadrant.
### Circle C:
- Center: [tex]\((3, -4)\)[/tex]
- Radius: [tex]\(1\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(3 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-4 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -4 + 1 = -3 < 0\)[/tex] (satisfies [tex]\(y + \text{radius} < 0\)[/tex] check)
- [tex]\(x - \text{radius} = 3 - 1 = 2 > 0\)[/tex] (satisfies [tex]\(x - \text{radius} > 0\)[/tex] check)
Circle C lies completely within the fourth quadrant.
### Circle D:
- Center: [tex]\((4, -2)\)[/tex]
- Radius: [tex]\(\sqrt{32} \approx 5.66\)[/tex]
- Check if it is within the fourth quadrant:
- [tex]\(x\)[/tex]-coordinate: [tex]\(4 > 0\)[/tex] (satisfies [tex]\(x > 0\)[/tex])
- [tex]\(y\)[/tex]-coordinate: [tex]\(-2 < 0\)[/tex] (satisfies [tex]\(y < 0\)[/tex])
- [tex]\(y + \text{radius} = -2 + 5.66 > 0\)[/tex] (fails [tex]\(y + \text{radius} < 0\)[/tex] check)
Circle D does not lie completely within the fourth quadrant.
Summary:
The circle that lies completely within the fourth quadrant is:
- Circle C: [tex]\((x-3)^2 + (y+4)^2 = 1\)[/tex]
Therefore, the correct answer is only Circle C, hence we submit "C."
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
