Understanding the Equation of a Sphere

The given equation of a sphere is \(x^2 + y^2 + z^2 - x + z - 2 = 0\). We need to find its radius. To do this, we compare the given equation with the standard general form of the equation of a sphere.

The general equation of a sphere in three dimensions is given by:

\(x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0\)

In this standard form, the center of the sphere is at the point \((-u, -v, -w)\) and the radius of the sphere is given by the formula:

\(r = \sqrt{u^2 + v^2 + w^2 - d}\)

Finding Parameters from the Given Sphere Equation

Let's compare the given equation \(x^2 + y^2 + z^2 - x + z - 2 = 0\) with the general form \(x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0\).

By comparing the coefficients of \(x, y, z\), and the constant term, we can find the values of \(u, v, w\), and \(d\).

• Coefficient of \(x\): \(2u = -1 \Rightarrow u = -\frac{1}{2}\)
• Coefficient of \(y\): There is no \(y\) term, so \(2v = 0 \Rightarrow v = 0\)
• Coefficient of \(z\): \(2w = 1 \Rightarrow w = \frac{1}{2}\)
• Constant term: \(d = -2\)

Calculating the Radius of the Sphere

Now that we have the values of \(u, v, w\), and \(d\), we can use the formula for the radius of the sphere, \(r = \sqrt{u^2 + v^2 + w^2 - d}\), to calculate the radius of the given sphere.

Substitute the values we found:

\(u = -\frac{1}{2}\), \(v = 0\), \(w = \frac{1}{2}\), \(d = -2\)

\(r = \sqrt{\left(-\frac{1}{2}\right)^2 + (0)^2 + \left(\frac{1}{2}\right)^2 - (-2)}\)

\(r = \sqrt{\frac{1}{4} + 0 + \frac{1}{4} + 2}\)

Combine the fractions:

\(r = \sqrt{\frac{1}{4} + \frac{1}{4} + \frac{8}{4}}\)

\(r = \sqrt{\frac{1+1+8}{4}}\)

\(r = \sqrt{\frac{10}{4}}\)

Simplify the fraction inside the square root:

\(r = \sqrt{\frac{5}{2}}\)

Thus, the radius of the sphere is \(\sqrt{\frac{5}{2}}\).

Summary of Finding the Radius of Sphere

To find the radius of a sphere from its general equation \(x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0\):

• Identify the coefficients \(2u, 2v, 2w\) for the \(x, y, z\) terms and the constant term \(d\).
• Calculate \(u, v, w, d\).
• Use the radius formula \(r = \sqrt{u^2 + v^2 + w^2 - d}\).

Applying this method to the given equation \(x^2 + y^2 + z^2 - x + z - 2 = 0\), we successfully calculated the radius of sphere as \(\sqrt{\frac{5}{2}}\).

This confirms the process for determining the radius of sphere from its algebraic form.