If `(p)/(a) + (q)/(b) + (r )/(c )=1 " & " (a)/(p) + (b)/(q) + (c )/(r )=0` find `(p^(2))/(a^(2)) + (q^(2))/(b^(2)) + (r^(2))/(c^(2))`= ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let:
\[ X = \frac{p}{a}, \quad Y = \frac{q}{b}, \quad Z = \frac{r}{c} \]
From the problem statement, we know:
\[ X + Y + Z = 1 \]

### Step 2: Use the Second Equation
The second equation given is:
\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0 \]
We can rewrite this using our definitions of \(X\), \(Y\), and \(Z\):
\[ \frac{1}{X} + \frac{1}{Y} + \frac{1}{Z} = 0 \]
This can be rewritten as:
\[ \frac{YZ + ZX + XY}{XYZ} = 0 \]
Thus, we have:
\[ YZ + ZX + XY = 0 \]

### Step 3: Use the Identity for Squares
We can use the identity:
\[ (X + Y + Z)^2 = X^2 + Y^2 + Z^2 + 2(XY + YZ + ZX) \]
Substituting the known values:
\[ 1^2 = X^2 + Y^2 + Z^2 + 2(0) \]
This simplifies to:
\[ 1 = X^2 + Y^2 + Z^2 \]

### Step 4: Relate Back to the Original Variables
Recall that:
\[ X = \frac{p}{a}, \quad Y = \frac{q}{b}, \quad Z = \frac{r}{c} \]
Thus:
\[ X^2 = \left(\frac{p}{a}\right)^2, \quad Y^2 = \left(\frac{q}{b}\right)^2, \quad Z^2 = \left(\frac{r}{c}\right)^2 \]
So we can write:
\[ X^2 + Y^2 + Z^2 = \left(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2}\right) \]
Therefore, we have:
\[ \frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} = 1 \]

### Final Answer
Thus, the value of \(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2}\) is:
\[ \boxed{1} \]