Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's consider the equation of the circle given:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
We can convert this equation into the standard form of a circle equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] by completing the square.
Step-by-step process:
1. Recognize the general form:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
- First, isolate the [tex]\(x\)[/tex] terms: [tex]\(x^2 - 2x\)[/tex].
- To complete the square, add and subtract [tex]\((\frac{-2}{2})^2 = 1\)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 \][/tex]
3. Rewrite the equation with the completed square:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
4. Combine constants on one side:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \implies (x - 1)^2 + y^2 = 9 \][/tex]
Now, the equation is in standard form [tex]\((x - 1)^2 + y^2 = 9\)[/tex].
From this standard form, we can derive the properties of the circle:
- The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Now, let's evaluate the given statements:
1. The radius of the circle is 3 units.
- True, because [tex]\(r = \sqrt{9} = 3\)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True, because the center is at [tex]\((1, 0)\)[/tex] which means [tex]\(k = 0\)[/tex] (on the [tex]\(x\)[/tex]-axis).
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False, because the center is at [tex]\((1, 0)\)[/tex] which means [tex]\(h \neq 0\)[/tex] (not on the [tex]\(y\)[/tex]-axis).
4. The standard form of the equation is [tex]\((x-1)^2+y^2=3\)[/tex].
- False, as we determined, the correct standard form is [tex]\((x - 1)^2 + y^2 = 9\)[/tex].
5. The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
- True, because the radius of the circle [tex]\(x^2 + y^2 = 9\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Therefore, the three correct statements are:
- The radius of the circle is 3 units.
- The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
We can convert this equation into the standard form of a circle equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] by completing the square.
Step-by-step process:
1. Recognize the general form:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
- First, isolate the [tex]\(x\)[/tex] terms: [tex]\(x^2 - 2x\)[/tex].
- To complete the square, add and subtract [tex]\((\frac{-2}{2})^2 = 1\)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 \][/tex]
3. Rewrite the equation with the completed square:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
4. Combine constants on one side:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \implies (x - 1)^2 + y^2 = 9 \][/tex]
Now, the equation is in standard form [tex]\((x - 1)^2 + y^2 = 9\)[/tex].
From this standard form, we can derive the properties of the circle:
- The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Now, let's evaluate the given statements:
1. The radius of the circle is 3 units.
- True, because [tex]\(r = \sqrt{9} = 3\)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True, because the center is at [tex]\((1, 0)\)[/tex] which means [tex]\(k = 0\)[/tex] (on the [tex]\(x\)[/tex]-axis).
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False, because the center is at [tex]\((1, 0)\)[/tex] which means [tex]\(h \neq 0\)[/tex] (not on the [tex]\(y\)[/tex]-axis).
4. The standard form of the equation is [tex]\((x-1)^2+y^2=3\)[/tex].
- False, as we determined, the correct standard form is [tex]\((x - 1)^2 + y^2 = 9\)[/tex].
5. The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
- True, because the radius of the circle [tex]\(x^2 + y^2 = 9\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Therefore, the three correct statements are:
- The radius of the circle is 3 units.
- The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- The radius of this circle is the same as the radius of the circle whose equation is [tex]\(x^2 + y^2 = 9\)[/tex].
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
