In a triangle P Q R , R = / 2 . If tan ( P 2 ) and tan ( Q 2 ) are roots of a x 2 + b x + c = 0 , where a 0 , then which one is true?

Question

In a triangle $P Q R$ , $∠ R = π / 2$ . If $\tan (\frac{P}{2})$ and $\tan (\frac{Q}{2})$ are roots of $a x^{2} + b x + c = 0,$ where $a \neq 0$ , then which one is true?

Smart Achievers · Accepted Answer

(1)

$∠ R = \frac{π}{2} \Rightarrow P + Q = \frac{π}{2}$

$\Rightarrow \tan (\frac{P}{2} + \frac{Q}{2}) = \tan \frac{π}{4} \Rightarrow \frac{\tan \frac{P}{2} + \tan \frac{Q}{2}}{1 - \tan \frac{P}{2} \tan \frac{Q}{2}} = 1$

$\Rightarrow - \frac{b}{a} = 1 - \frac{c}{a}$ $∴$ $c = a + b$ $[∵ \tan \frac{P}{2}, \tan \frac{Q}{2} are roots of a x^{2} + b x + c = 0]$

$[∴ \tan \frac{P}{2} + \tan \frac{Q}{2} = - \frac{b}{a} and \tan \frac{P}{2} \cdot \tan \frac{Q}{2} = \frac{c}{a}]$

Solution

Q.
In a triangle $P Q R$ , $∠ R = π / 2$ . If $\tan (\frac{P}{2})$ and $\tan (\frac{Q}{2})$ are roots of $a x^{2} + b x + c = 0,$ where $a \neq 0$ , then which one is true

1 $c = a + b$

2 $a = b + c$

3 $b = a + c$

4 $b = c$

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Solution

Q. In a triangle PQR, ∠R=π/2. If tan(P2) and tan(Q2) are roots of ax2+bx+c=0, where a≠0, then which one is true

1 c=a+b

2 a=b+c

3 b=a+c

4 b=c