A plane passes through the fixed point (a, b, c) and cuts the coordinates axes respectively in A, B and C. The locus of the centre of the sphere passing through origin and A, B and C will be :

Question

Accepted Answer

To determine the locus of the center of the sphere passing through the origin and the points A, B, and C, we start by analyzing the given conditions.

1. **Plane Equation**: The plane passes through the fixed point $(a, b, c)$ and intersects the coordinate axes at points A, B, and C. The general equation of a plane that cuts the coordinate axes at points A, B, and C is given by:
   $$
   \frac{x}{x_1} + \frac{y}{y_1} + \frac{z}{z_1} = 1
   $$
   where $x_1$, $y_1$, and $z_1$ are the intercepts on the x, y, and z axes, respectively.

2. **Sphere Center**: The sphere passes through the origin and the points A, B, and C. The center of such a sphere is the midpoint of the line segment joining the origin to the point $(x_1, y_1, z_1)$.

3. **Midpoint Calculation**: The midpoint of the line segment from the origin $(0, 0, 0)$ to $(x_1, y_1, z_1)$ is:
   $$
   \left(\frac{x_1}{2}, \frac{y_1}{2}, \frac{z_1}{2}\right)
   $$

4. **Locus of the Center**: The locus of the center of the sphere is determined by the condition that the plane passes through the fixed point $(a, b, c)$. Substituting this point into the plane equation:
   $$
   \frac{a}{x_1} + \frac{b}{y_1} + \frac{c}{z_1} = 1
   $$

5. **Correct Option**: The equation $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2$ represents the locus of the center of the sphere. This is because the center of the sphere is at $\left(\frac{x_1}{2}, \frac{y_1}{2}, \frac{z_1}{2}\right)$, and substituting these into the plane equation gives:
   $$
   \frac{a}{\frac{x_1}{2}} + \frac{b}{\frac{y_1}{2}} + \frac{c}{\frac{z_1}{2}} = 2 \left(\frac{a}{x_1} + \frac{b}{y_1} + \frac{c}{z_1}\right) = 2 \times 1 = 2
   $$

6. **Explanation of Incorrect Options**:
   - **Option 1**: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ does not account for the scaling factor of 2 that arises from the midpoint calculation.
   - **Option 2**: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 2$ is incorrect because it does not match the form derived from the midpoint substitution.
   - **Option 3**: $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1$ is incorrect because it does not reflect the doubling effect from the midpoint calculation.

Therefore, the correct answer is Option 4: $\frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2$, as it accurately represents the locus of the center of the sphere passing through the origin and the points A, B, and C.