If `p/a + q/b + r/c = 1` and `a/p + b/q + c/r = 0`, where p, q,r and a, b,c are non-zero, then the value of `(p^2)/(a^2) + (q^2)/(b^2) + (r^2)/(c^2)` is

Options

A. (a) `-1`
B. (b) `0`
C. (c) `1`
D. (d) `2`

Explanation

To solve the problem, we start with the given equations:

1. \(\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1\)
2. \(\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0\)

We need to find the value of \(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2}\).

### Step 1: Square the first equation
We start by squaring the first equation:

\[ \left(\frac{p}{a} + \frac{q}{b} + \frac{r}{c}\right)^2 = 1^2 \]

### Step 2: Expand the squared equation
Using the formula \((x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz)\), we expand:

\[ \frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} + 2\left(\frac{pq}{ab} + \frac{qr}{bc} + \frac{rp}{ca}\right) = 1 \]

### Step 3: Substitute the second equation
From the second equation, we know:

\[ \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0 \]

This can be rewritten as:

\[ \frac{bc}{pq} + \frac{ca}{rp} + \frac{ab}{qr} = 0 \]

### Step 4: Rearranging the second equation
Multiplying through by \(pqr\) gives:

\[ bc \cdot r + ca \cdot p + ab \cdot q = 0 \]

This implies:

\[ \frac{pq}{ab} + \frac{qr}{bc} + \frac{rp}{ca} = 0 \]

### Step 5: Substitute back into the expanded equation
Substituting this back into our expanded equation gives:

\[ \frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} + 2 \cdot 0 = 1 \]

Thus, we have:

\[ \frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} = 1 \]

### Final Result
The value of \(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2}\) is:

\[ \boxed{1} \]