Understanding the Problem: Locus of Sphere Center

The question asks for the locus of the center of a sphere that passes through four specific points: the origin O, and the points A, B, and C where a variable plane intersects the x, y, and z axes respectively. The variable plane has a constraint: it always passes through a fixed point \((a, b, c)\).

Defining the Variable Plane and Intercepts

Let the equation of the variable plane be represented by its intercepts on the coordinate axes. If the plane cuts the x-axis at A, the y-axis at B, and the z-axis at C, their coordinates can be denoted as:

• Point A: \((p, 0, 0)\) (x-intercept is \(p\))
• Point B: \((0, q, 0)\) (y-intercept is \(q\))
• Point C: \((0, 0, r)\) (z-intercept is \(r\))

The origin is point O \((0, 0, 0)\).

The equation of a plane with intercepts \(p, q, r\) on the x, y, and z axes respectively is given by:

\[ \frac{X}{p} + \frac{Y}{q} + \frac{Z}{r} = 1 \]

Applying the Fixed Point Constraint

We are given that the variable plane passes through the fixed point \((a, b, c)\). This means that the coordinates \((a, b, c)\) must satisfy the plane equation. Substituting \((a, b, c)\) into the equation of the plane:

\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 1 \]

This equation gives us a crucial relationship between the intercepts \(p, q, r\) as the plane varies but always passes through \((a, b, c)\).

Finding the Sphere Passing Through O, A, B, C

The sphere passes through the points O \((0, 0, 0)\), A \((p, 0, 0)\), B \((0, q, 0)\), and C \((0, 0, r)\). The general equation of a sphere is:

\[ X^2 + Y^2 + Z^2 + 2UX + 2VY + 2WZ + D = 0 \]

Let's use the given points to find the values of \(U, V, W, D\).

• Since the sphere passes through the origin O \((0, 0, 0)\):
  \(0^2 + 0^2 + 0^2 + 2U(0) + 2V(0) + 2W(0) + D = 0\)
  This gives us \(D = 0\).
• The sphere passes through A \((p, 0, 0)\):
  \(p^2 + 0^2 + 0^2 + 2U(p) + 2V(0) + 2W(0) + 0 = 0\)
  \(p^2 + 2Up = 0\). Assuming \(p \neq 0\), we can divide by \(p\): \(p + 2U = 0\), so \(U = -\frac{p}{2}\).
• The sphere passes through B \((0, q, 0)\):
  \(0^2 + q^2 + 0^2 + 2U(0) + 2V(q) + 2W(0) + 0 = 0\)
  \(q^2 + 2Vq = 0\). Assuming \(q \neq 0\), \(q + 2V = 0\), so \(V = -\frac{q}{2}\).
• The sphere passes through C \((0, 0, r)\):
  \(0^2 + 0^2 + r^2 + 2U(0) + 2V(0) + 2W(r) + 0 = 0\)
  \(r^2 + 2Wr = 0\). Assuming \(r \neq 0\), \(r + 2W = 0\), so \(W = -\frac{r}{2}\).

The center of the sphere \(X^2 + Y^2 + Z^2 + 2UX + 2VY + 2WZ + D = 0\) is given by the coordinates \((-U, -V, -W)\).

Substituting the values we found for \(U, V, W\):

Center of the sphere \(= \left(-\left(-\frac{p}{2}\right), -\left(-\frac{q}{2}\right), -\left(-\frac{r}{2}\right)\right) = \left(\frac{p}{2}, \frac{q}{2}, \frac{r}{2}\right)\)

Determining the Locus of the Center

Let the coordinates of the center of the sphere be \((x, y, z)\). So, we have:

\[ x = \frac{p}{2}, \quad y = \frac{q}{2}, \quad z = \frac{r}{2} \]

From these equations, we can express the intercepts \(p, q, r\) in terms of the coordinates of the center \((x, y, z)\):

\[ p = 2x, \quad q = 2y, \quad r = 2z \]

Now, we use the constraint equation we found earlier that relates the intercepts \(p, q, r\):

\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 1 \]

Substitute the expressions for \(p, q, r\) in terms of \(x, y, z\) into this equation:

\[ \frac{a}{2x} + \frac{b}{2y} + \frac{c}{2z} = 1 \]

To simplify, multiply the entire equation by 2:

\[ 2 \left( \frac{a}{2x} + \frac{b}{2y} + \frac{c}{2z} \right) = 2(1) \]

\[ \frac{2a}{2x} + \frac{2b}{2y} + \frac{2c}{2z} = 2 \]

\[ \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2 \]

This is the equation that the coordinates \((x, y, z)\) of the center of the sphere must satisfy. Therefore, this equation represents the locus of the center of the sphere OABC.

Conclusion and Matching Option

The locus of the center of the sphere OABC is given by the equation \(\frac{{\rm{a}}}{{\rm{x}}} + \frac{{\rm{b}}}{{\rm{y}}} + \frac{{\rm{c}}}{{\rm{z}}} = 2\).

Comparing this result with the given options, we find that it matches option 3.

Revision Table: Key Concepts in Locus Problems

Additional Information: Exploring Related Concepts

Locus problems in 3D geometry often involve finding relationships between the coordinates of a moving point based on geometric constraints. In this problem, the "moving point" is the center of the sphere, whose position depends on the intercepts \(p, q, r\). The intercepts themselves are related because the plane must pass through the fixed point \((a, b, c)\).

The fact that the sphere passes through the origin and three points on the axes simplifies finding the sphere's equation and center considerably. The points \((p, 0, 0)\), \((0, q, 0)\), \((0, 0, r)\) define a cuboid with opposite vertices at the origin and \((p, q, r)\). A sphere passing through O, A, B, C (and also the other three points \((p, q, 0), (p, 0, r), (0, q, r)\) and \((p, q, r)\)) has its center at the midpoint of the diagonal from the origin to \((p, q, r)\), which is \((p/2, q/2, r/2)\). This confirms our finding for the center's coordinates.