Step 1: Understanding the Concept:
We are looking for the locus (path) of the center of a sphere that is defined by four points: the origin O and the intercepts A, B, C of a variable plane. This variable plane has the constraint that it always passes through a fixed point (a,b,c).
Step 2: Key Formula or Approach:
1. Let the equation of the plane be in intercept form: \(\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1\). The intercepts are \(A=(p,0,0)\), \(B=(0,q,0)\), \(C=(0,0,r)\).
2. The sphere passes through O(0,0,0), A, B, and C. The equation of such a sphere is \(x^2+y^2+z^2 - px - qy - rz = 0\).
3. The center of this sphere is \((\frac{p}{2}, \frac{q}{2}, \frac{r}{2})\).
4. The plane passes through the fixed point (a,b,c). Substitute this into the plane equation to get a condition on p, q, r.
5. Use the coordinates of the center to relate p, q, r to x, y, z of the locus. Substitute this into the condition from step 4 to find the locus equation.
Step 3: Detailed Explanation:
Let the variable plane cut the axes at \(A(p,0,0)\), \(B(0,q,0)\), and \(C(0,0,r)\). The equation of this plane in intercept form is:
\[ \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 \]
Since this plane passes through the fixed point \((a,b,c)\), these coordinates must satisfy the plane's equation:
\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 1 \quad (). \]
Now consider the sphere OABC. The equation of a sphere passing through the origin is \(x^2+y^2+z^2+2ux+2vy+2wz=0\).
Since it passes through A(p,0,0), B(0,q,0), and C(0,0,r), we find \(2u=-p, 2v=-q, 2w=-r\).
The equation of the sphere is \(x^2+y^2+z^2 - px - qy - rz = 0\).
The center of this sphere is \((-\frac{-p}{2}, -\frac{-q}{2}, -\frac{-r}{2}) = (\frac{p}{2}, \frac{q}{2}, \frac{r}{2})\).
Let the coordinates of the center be \((x_c, y_c, z_c)\). To find its locus, we will just use \((x, y, z)\).
\[ x = \frac{p}{2} \implies p = 2x \]
\[ y = \frac{q}{2} \implies q = 2y \]
\[ z = \frac{r}{2} \implies r = 2z \]
Now, we substitute these expressions for p, q, and r into the condition we found earlier (\(\)):
MATH_cae518a31add429abcfb8a7671f5ac05
Multiply the entire equation by 2:
MATH_994e15a07100467eb432ac14b8191c2d
This is the equation of the locus of the center of the sphere.
Step 4: Final Answer:
The locus of the centre of the sphere OABC is \(\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2\), which corresponds to option (B).