In a triangle PQR angle R=90 .If tan P/2 and tan Q/2 are the roots of ax^2 +bx+c=0,a is not equal to 0 then find the relation between a,b and c.

urgent

Answer :
Given
tan$\frac{P}{2}$  And tan$\frac{Q}{2}$  are the roots of ax² + bx + c = 0
So by relationship between zeros and coefficient , we get

tan$\frac{P}{2}$  + tan$\frac{Q}{2}$ = $-\frac{b}{a}$          -------------- ( 1 )
And
tan$\frac{P}{2}$ tan$\frac{Q}{2}$ = $\frac{c}{a}$                  --------------- ( 2 )
In triangle PQR 
 
So, we know sum of all internal angles in triangle = 180

90 + P + Q = 180 

P + Q = 90

Divide by 2 on both hand side , we get

$\Rightarrow$ $\frac{P}{2}$ + $\frac{Q}{2}$  = 45
Taking tan on both hand side and get

$\Rightarrow$tan ( $\frac{P}{2}$ + $\frac{Q}{2}$ ) = tan 45
 
 $\Rightarrow$$\frac{\tan {\displaystyle \frac{P}{2}} + \tan {\displaystyle \frac{Q}{2}}}{1 - \tan \frac{P}{2}\tan \frac{Q}{2}}$  = 1                 ( As we know tan 45 = 1 )

Substitute values from equation 1 and 2 , and get

$\Rightarrow$$\frac{-{\displaystyle \frac{b}{a}}}{1 -{\displaystyle \frac{c}{a}}}$  = 1 

$\Rightarrow$- b  = a  - c

$\Rightarrow$ c  - a  = b                                   ( Ans )