Let ( ) and ( ) be the roots of the equation (p x^2+q x+r=0, p 0 ). If (p, q, r ) are in AP and ( 1 + 1 =4 ), then the value of (| - | ) is

Step-by-step solution

Since, ( ) and ( ) are roots of quadratic equation (p x^2+q x+r=0, p 0 ) and (p, q, r ) are in AP such that ( 1 + 1 =4 ) ...(i) Let the (p=q-d ) and (r=q+d ), where (d ) is the common difference of AP ( ) Sum of the roots ( + = -q q-d ) and product of the roots ( = q+d q-d ) ( ) from Eq. (i), we have ( + =4 -q q+d =4 5 q+4 d=0 ) ...(ii) Now, (| - |= ( + )^2-4 ) ( aligned & = q^2 (q-d)^2 -4 q+d q-d = q^2-4 (q^2-d^2 ) (q-d)^2 & = 4 d^2-3 q^2 (q-d)^2 = 4 (- 5 4 q )^2-3 q^2 (q+ 5 4 q )^2 & = 25 4 -3 ( 9 4 )^2 = 2 9 13