Multiple choice

In a triangle PQR, $\angle R$ = $\displaystyle \frac{\pi }{2}$. If tan $\displaystyle \frac{p}{2}$ and tan $\displaystyle \frac{Q }{2}$ are the roots of the equation $ax^{2}+bx+c = 0 (a\neq 0)$ then

  1. $a+b=c$
  2. $b+c=a$
  3. $a+c=b$
  4. $b=c$
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A Correct answer
Explanation

In a triangle, P+Q+R = 180. Since R=90, P+Q=90, so P/2 + Q/2 = 45. tan(P/2 + Q/2) = tan(45) = 1. Using the formula tan(A+B) = (tan A + tan B) / (1 - tan A tan B), we get (sum of roots) / (1 - product of roots) = 1. Sum = -b/a, Product = c/a. (-b/a) / (1 - c/a) = 1. -b / (a-c) = 1. -b = a - c, so a + b = c.

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