Question : The value of $\frac{p^{2}- (q - r)^{2}}{(p + r)^{2} - (q)^{2}}$ + $\frac{q^{2} - (p - r)^{2}}{(p + q)^{2} - (r)^{2}}$ + $\frac{r^{2}- (p - q)^{2}}{(q + r)^{2} - (p)^{2}}$ is:
Option 1: 1
Option 2: 2
Option 3: 0
Option 4: 3
Correct Answer: 1
Solution :
$\frac{p^{2} - (q - r)^{2}}{(p + r)^{2} - (q)^{2}}$ + $\frac{q^{2} - (p - r)^{2}}{(p + q)^{2} - (r)^{2}}$ + $\frac{r^{2} - (p – q)^{2}}{(q + r)^{2} - (p)^{2}}$
= $\frac{(p + q - r)(p - q + r)}{(p + q + r)(p - q + r)} + \frac{(p + q - r)(-p + q + r)}{(p + q + r)(p + q - r)} + \frac{(p - q + r)(–p+ q + r)}{(p + q + r)(–p + q + r)}$
= $\frac{p + q - r - p + q + r + p -q + r}{p + q+ r}$
= $\frac{p + q + r}{p + q + r}$
= $1$
Hence, the correct answer is 1.
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