If `a x+b y+c z=p` , then minimum value of `x^2+y^2+z^2` is `(p/(a+b+c))^2` (b) `(p^2)/(a^2+b^2+c^2)` `(a^2+b^2+c^2)/(p^2)` (d) `((a+b+c)/p)^2`

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 equation \( ax + by + cz = p \), we can use the method of Lagrange multipliers or geometric interpretation. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We are given a linear equation in three variables \( ax + by + cz = p \) and we need to minimize the expression \( x^2 + y^2 + z^2 \).

### Step 2: Set Up the Distance
The expression \( x^2 + y^2 + z^2 \) represents the square of the distance from the origin (0, 0, 0) to the point (x, y, z). We want to find the point on the plane defined by \( ax + by + cz = p \) that is closest to the origin.

### Step 3: Use the Distance Formula
The distance \( d \) from the origin to the plane can be calculated using the formula for the distance from a point to a plane. The distance \( d \) from the point (0, 0, 0) to the plane \( ax + by + cz = p \) is given by:
\[ d = \frac{|ax_0 + by_0 + cz_0 - p|}{\sqrt{a^2 + b^2 + c^2}} \]
where \( (x_0, y_0, z_0) \) is the point (0, 0, 0).

### Step 4: Substitute the Values
Substituting \( x_0 = 0 \), \( y_0 = 0 \), and \( z_0 = 0 \) into the distance formula gives:
\[ d = \frac{|0 + 0 + 0 - p|}{\sqrt{a^2 + b^2 + c^2}} = \frac{| - p |}{\sqrt{a^2 + b^2 + c^2}} = \frac{p}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 5: Find the Minimum Value of \( x^2 + y^2 + z^2 \)
Since \( d^2 = x^2 + y^2 + z^2 \), we have:
\[ x^2 + y^2 + z^2 = d^2 = \left( \frac{p}{\sqrt{a^2 + b^2 + c^2}} \right)^2 = \frac{p^2}{a^2 + b^2 + c^2} \]

### Conclusion
Thus, the minimum value of \( x^2 + y^2 + z^2 \) given the constraint \( ax + by + cz = p \) is:
\[ \frac{p^2}{a^2 + b^2 + c^2} \]

### Final Answer
The correct option is (b) \( \frac{p^2}{a^2 + b^2 + c^2} \).

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