`2x^(2) + 3x - alpha - 0 " has roots "-2 and beta " while the equation "x^(2) - 3mx + 2m^(2) = 0 " has both roots positive, where " alpha gt 0 and beta gt 0.`
`Delta PQR, angleR=pi/2. " If than "(P/2) and tan (Q/2) " are the roots of the equation " ax^(2) + bx + c = 0 ,` then which one of the following is correct ?

Options

A. `a=b+c`
B. `b=c+a`
C. `c=a+b`
D. `b=c`

Answer

C

Explanation

`angleR=(pi)/2`

` :. angle P+angleQ= (pi)/2`
`rArr (angleP+angleQ)/2=(pi)/4 " "`…(1)
`tan.(p/2+Q/2) = (tan.P/2+tan.Q/2)/(1-tan.P/2tan.Q/2)" "`…(2)
Given , `tan.P/2andtan.Q/2 " are roots of " ax^(2) + bx + c = 0 `.
` :. tan.P/2 + tan.Q/2=(-b)/2,tan.P/2 .tan Q/2 = c/a.`
` :. (2) rArr tan.(P/2+Q/2) = (-b/a)/(1-c/a)`
` rArr tan. (pi)/4 = ((-b)/a)/(1-c/a) " "` (from (1))
`rArr 1= -b/(a-c) rArr -b = a - c rArra+b=c`.