Find the value if `(p^(2)+q^(2))/(r^(2)+s^(2))` if `p:q: : r:s`

Options

A. `1/4`
B. `(pq)/(rs)`
C. `((p+q)/(r+s))^(2)`
D. both b and c

Answer

To solve the problem of finding the value of \((p^2 + q^2)/(r^2 + s^2)\) given that \(p:q::r:s\), we can follow these steps:

### Step 1: Set up the ratio
Since \(p:q::r:s\), we can express this as:
\[ \frac{p}{q} = \frac{r}{s} \]
Let’s denote this common ratio as \(k\). Therefore, we can write:
\[ p = kq \quad \text{and} \quad r = ks \]

### Step 2: Substitute the values into the expression
Now we substitute \(p\) and \(r\) in terms of \(q\) and \(s\) into the expression \((p^2 + q^2)/(r^2 + s^2)\):
\[ \frac{(kq)^2 + q^2}{(ks)^2 + s^2} \]

### Step 3: Simplify the expression
Now we simplify the numerator and the denominator:
- The numerator becomes:
\[ (k^2q^2 + q^2) = q^2(k^2 + 1) \]
- The denominator becomes:
\[ (k^2s^2 + s^2) = s^2(k^2 + 1) \]

### Step 4: Combine the fractions
Now we can rewrite the expression:
\[ \frac{q^2(k^2 + 1)}{s^2(k^2 + 1)} \]

### Step 5: Cancel out the common terms
Since \(k^2 + 1\) is common in both the numerator and the denominator, we can cancel it out (assuming \(k^2 + 1 \neq 0\)):
\[ \frac{q^2}{s^2} \]

### Step 6: Express the final result
Thus, we have:
\[ \frac{p^2 + q^2}{r^2 + s^2} = \frac{q^2}{s^2} \]

### Conclusion
The final value of \(\frac{p^2 + q^2}{r^2 + s^2}\) is \(\left(\frac{q}{s}\right)^2\).

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