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The problem is asking to find the equation of the sphere for which the given circle is a great circle.

The equations given in the problem are:

1. \( x^2 + y^2 + z^2 + y - 2x + 2 = 0 \) (this is the equation of the sphere),
2. \( 2x + 3y + 4z = 8 \) (this is the equation of the plane containing the circle).

### Step-by-step Solution:

#### Step 1: Rewrite the sphere equation in standard form
The equation of the sphere is:

\[
x^2 + y^2 + z^2 + y - 2x + 2 = 0
\]

Group terms related to \(x\), \(y\), and \(z\), and complete the square for each variable:

- For \(x^2 - 2x\), complete the square:
  \[
  x^2 - 2x = (x - 1)^2 - 1
  \]

- For \(y^2 + y\), complete the square:
  \[
  y^2 + y = \left(y + \frac{1}{2}\right)^2 - \frac{1}{4}
  \]

- For \(z^2\), it's already a perfect square:
  \[
  z^2 = z^2
  \]

Now substitute these back into the equation:

\[
(x - 1)^2 - 1 + \left(y + \frac{1}{2}\right)^2 - \frac{1}{4} + z^2 + 2 = 0
\]

Simplify:

\[
(x - 1)^2 + \left(y + \frac{1}{2}\right)^2 + z^2 = \frac{1}{4}
\]

Thus, the equation of the sphere is:

\[
(x - 1)^2 + \left(y + \frac{1}{2}\right)^2 + z^2 = \left(\frac{1}{2}\right)^2
\]

This represents a sphere with center \((1, -\frac{1}{2}, 0)\) and radius \(\frac{1}{2}\).

#### Step 2: Verify the great circle condition
A great circle is the intersection of a sphere and a plane that

Discover how to find the equation of a sphere for which a given circle is a great circle. Step-by-step solution includes completing the square and verifying conditions. Suitable for advanced mathematics students.

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