Answer :
To find the radius of the circle given by the equation [tex]\( x^2 + y^2 + 4x + 8y - 10 = 0 \)[/tex], we need to rewrite the equation in the standard form of a circle, which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
### Step-by-Step Solution
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[ x^2 + 4x + y^2 + 8y - 10 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
For [tex]\(x^2 + 4x\)[/tex], we take the coefficient of [tex]\(x\)[/tex], which is 4, divide it by 2 (giving us 2), and then square it (resulting in 4). Add and subtract 4 within the equation:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
For [tex]\(y^2 + 8y\)[/tex], we take the coefficient of [tex]\(y\)[/tex], which is 8, divide it by 2 (giving us 4), and then square it (resulting in 16). Add and subtract 16 within the equation:
[tex]\[ y^2 + 8y = (y + 4)^2 - 16 \][/tex]
4. Substitute these completed squares back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y + 4)^2 - 16 - 10 = 0 \][/tex]
5. Combine the constants on the left-hand side and move them to the right-hand side:
[tex]\[ (x + 2)^2 + (y + 4)^2 - 30 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y + 4)^2 = 30 \][/tex]
6. Identify the standard form of the circle equation:
The given equation [tex]\((x + 2)^2 + (y + 4)^2 = 30\)[/tex] is now in the standard form [tex]\((x - (-2))^2 + (y - (-4))^2 = r^2\)[/tex].
We can see that the radius squared ([tex]\(r^2\)[/tex]) is equal to 30.
7. Find the radius:
[tex]\[ r = \sqrt{30} \][/tex]
8. Round the radius to the nearest thousandth:
[tex]\[ r \approx 5.477 \][/tex]
Therefore, the radius of the circle, rounded to the nearest thousandth, is approximately [tex]\( 5.477 \)[/tex].
### Step-by-Step Solution
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[ x^2 + 4x + y^2 + 8y - 10 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
For [tex]\(x^2 + 4x\)[/tex], we take the coefficient of [tex]\(x\)[/tex], which is 4, divide it by 2 (giving us 2), and then square it (resulting in 4). Add and subtract 4 within the equation:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
For [tex]\(y^2 + 8y\)[/tex], we take the coefficient of [tex]\(y\)[/tex], which is 8, divide it by 2 (giving us 4), and then square it (resulting in 16). Add and subtract 16 within the equation:
[tex]\[ y^2 + 8y = (y + 4)^2 - 16 \][/tex]
4. Substitute these completed squares back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y + 4)^2 - 16 - 10 = 0 \][/tex]
5. Combine the constants on the left-hand side and move them to the right-hand side:
[tex]\[ (x + 2)^2 + (y + 4)^2 - 30 = 0 \][/tex]
[tex]\[ (x + 2)^2 + (y + 4)^2 = 30 \][/tex]
6. Identify the standard form of the circle equation:
The given equation [tex]\((x + 2)^2 + (y + 4)^2 = 30\)[/tex] is now in the standard form [tex]\((x - (-2))^2 + (y - (-4))^2 = r^2\)[/tex].
We can see that the radius squared ([tex]\(r^2\)[/tex]) is equal to 30.
7. Find the radius:
[tex]\[ r = \sqrt{30} \][/tex]
8. Round the radius to the nearest thousandth:
[tex]\[ r \approx 5.477 \][/tex]
Therefore, the radius of the circle, rounded to the nearest thousandth, is approximately [tex]\( 5.477 \)[/tex].
