Join the IDNLearn.com community and start finding the answers you need today. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
To determine which point lies on the circle given by the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], let's analyze the situation step by step.
1. Understand the Circle Equation:
- The standard form of a circle's equation is [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
- From the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we can identify:
- The center [tex]\((h, k)\)[/tex] of the circle is [tex]\((3, -4)\)[/tex].
- The radius [tex]\(r\)[/tex] of the circle is [tex]\(6\)[/tex].
2. Calculate the Distances from the Center to Each Point:
- The distance [tex]\(d\)[/tex] from the center [tex]\((3, -4)\)[/tex] to a point [tex]\((x, y)\)[/tex] is given by the formula:
[tex]\[ d = \sqrt{(x - 3)^2 + (y + 4)^2} \][/tex]
3. Check Each Point:
- For point [tex]\((9, -2)\)[/tex]:
[tex]\[ d = \sqrt{(9 - 3)^2 + (-2 + 4)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \][/tex]
- The distance is approximately 6.32, not equal to the radius 6.
- For point [tex]\((0, 11)\)[/tex]:
[tex]\[ d = \sqrt{(0 - 3)^2 + (11 + 4)^2} = \sqrt{(-3)^2 + 15^2} = \sqrt{9 + 225} = \sqrt{234} \approx 15.3 \][/tex]
- The distance is approximately 15.3, not equal to the radius 6.
- For point [tex]\((3, 10)\)[/tex]:
[tex]\[ d = \sqrt{(3 - 3)^2 + (10 + 4)^2} = \sqrt{0^2 + 14^2} = \sqrt{0 + 196} = \sqrt{196} = 14 \][/tex]
- The distance is 14, not equal to the radius 6.
- For point [tex]\(( - 9, 4\)[/tex]):
[tex]\[ d = \sqrt{(-9 - 3)^2 + (4 + 4)^2} = \sqrt{(-12)^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} \approx 14.42 \][/tex]
- The distance is approximately 14.42, not equal to the radius 6.
- For point [tex]\((-3, -4)\)[/tex]:
[tex]\[ d = \sqrt{(-3 - 3)^2 + (-4 + 4)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36 + 0} = \sqrt{36} = 6 \][/tex]
- The distance is exactly 6, which is equal to the radius of the circle.
4. Conclusion:
- The only point whose distance from the center [tex]\((3, -4)\)[/tex] is exactly equal to the radius [tex]\(6\)[/tex] is [tex]\((-3, -4)\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{E. (-3, -4)} \][/tex]
1. Understand the Circle Equation:
- The standard form of a circle's equation is [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
- From the equation [tex]\((x-3)^2 + (y+4)^2 = 6^2\)[/tex], we can identify:
- The center [tex]\((h, k)\)[/tex] of the circle is [tex]\((3, -4)\)[/tex].
- The radius [tex]\(r\)[/tex] of the circle is [tex]\(6\)[/tex].
2. Calculate the Distances from the Center to Each Point:
- The distance [tex]\(d\)[/tex] from the center [tex]\((3, -4)\)[/tex] to a point [tex]\((x, y)\)[/tex] is given by the formula:
[tex]\[ d = \sqrt{(x - 3)^2 + (y + 4)^2} \][/tex]
3. Check Each Point:
- For point [tex]\((9, -2)\)[/tex]:
[tex]\[ d = \sqrt{(9 - 3)^2 + (-2 + 4)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \][/tex]
- The distance is approximately 6.32, not equal to the radius 6.
- For point [tex]\((0, 11)\)[/tex]:
[tex]\[ d = \sqrt{(0 - 3)^2 + (11 + 4)^2} = \sqrt{(-3)^2 + 15^2} = \sqrt{9 + 225} = \sqrt{234} \approx 15.3 \][/tex]
- The distance is approximately 15.3, not equal to the radius 6.
- For point [tex]\((3, 10)\)[/tex]:
[tex]\[ d = \sqrt{(3 - 3)^2 + (10 + 4)^2} = \sqrt{0^2 + 14^2} = \sqrt{0 + 196} = \sqrt{196} = 14 \][/tex]
- The distance is 14, not equal to the radius 6.
- For point [tex]\(( - 9, 4\)[/tex]):
[tex]\[ d = \sqrt{(-9 - 3)^2 + (4 + 4)^2} = \sqrt{(-12)^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} \approx 14.42 \][/tex]
- The distance is approximately 14.42, not equal to the radius 6.
- For point [tex]\((-3, -4)\)[/tex]:
[tex]\[ d = \sqrt{(-3 - 3)^2 + (-4 + 4)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36 + 0} = \sqrt{36} = 6 \][/tex]
- The distance is exactly 6, which is equal to the radius of the circle.
4. Conclusion:
- The only point whose distance from the center [tex]\((3, -4)\)[/tex] is exactly equal to the radius [tex]\(6\)[/tex] is [tex]\((-3, -4)\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{E. (-3, -4)} \][/tex]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
