If `ax+by+cz = p`, then minimum value of `x^2 + y^2 + z^2` is

Options

A. `((p)/(a+b+c))^(2)`
B. `(p^(2))/(a^(2)+b^(2)+c^(2))`
C. `(a^(2)+b^(2)+c^(2))/(p^(2))`
D. 0

Answer

To find the minimum value of \( x^2 + y^2 + z^2 \) given the constraint \( ax + by + cz = p \), we can use the method of Lagrange multipliers or geometric interpretation. Here, we will use a geometric approach.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to minimize the expression \( x^2 + y^2 + z^2 \) subject to the constraint \( ax + by + cz = p \). This expression represents the square of the distance from the point \( (x, y, z) \) to the origin \( (0, 0, 0) \).

2. **Distance Formula**: The distance \( d \) from the point \( (x, y, z) \) to the origin is given by:
\[ d = \sqrt{x^2 + y^2 + z^2} \]
Therefore, we want to minimize \( d^2 = x^2 + y^2 + z^2 \).

3. **Geometric Interpretation**: The equation \( ax + by + cz = p \) represents a plane in three-dimensional space. The minimum distance from the origin to this plane occurs when the line connecting the origin to the plane is perpendicular to the plane.

4. **Distance from the Origin to the Plane**: The formula for the distance \( D \) from a point \( (x_0, y_0, z_0) \) to the plane \( ax + by + cz = d \) is given by:
\[ D = \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}} \]
In our case, the point is the origin \( (0, 0, 0) \) and \( d = p \):
\[ D = \frac{|0 + 0 + 0 - p|}{\sqrt{a^2 + b^2 + c^2}} = \frac{|p|}{\sqrt{a^2 + b^2 + c^2}} \]

5. **Finding the Minimum Value**: The minimum value of \( d^2 \) is then:
\[ d^2 = D^2 = \left(\frac{|p|}{\sqrt{a^2 + b^2 + c^2}}\right)^2 = \frac{p^2}{a^2 + b^2 + c^2} \]

6. **Conclusion**: Therefore, the minimum value of \( x^2 + y^2 + z^2 \) subject to the constraint \( ax + by + cz = p \) is:
\[ \frac{p^2}{a^2 + b^2 + c^2} \]

### Final Answer:
The minimum value of \( x^2 + y^2 + z^2 \) is \( \frac{p^2}{a^2 + b^2 + c^2} \).