Question

let `alpha ,beta` be roots of `ax^2+bx+c=0` and `gamma,delta` be the roots of `px^2+qx+r=0`and `D_1` and `D_2` be the respective equations .if `alpha,beta,gamma,delta` in `A.P.` then `D_1/D_2` is
A. `(a^(2))/(b^(2))`
B. `(a^(2))/(p^(2))`
C. `(b^(2))/(q^(2))`
D. `(c^(2))/(r^(2))`

Answer 1

Correct Answer - B
We have, `alpha+beta = -(b)/(a), alpha beta = (c)/(a), gamma + delta = -(q)/(p) and gamma delta = (r)/(p)`
`D_(1) = b^(2) - 4ac and D_(2) = q^(2) - 4 pr`
Now, `alpha, beta, gamma, delta` are in A.P.
`rArr" "beta-alpha = delta - gamma`
`rArr" "(beta - alpha)^(2) = (delta - gamma)^(2)`
`rArr" "(beta + alpha)^(2) - 4 alpha beta = (gamma + delta)^(2) - 4 gamma delta`
`rArr" "(b^(2))/(a^(2))-(4c)/(a) = (q^(2))/(p^(2)) - (4r)/(p)`
`rArr" "(b^(2)-4ac)/(a^(2))=(q^(2)-4rp)/(p^(2)) rArr (D_(1))/(a^(2))=(D_(2))/(p^(2)) rArr D_(1))/(D_(2))=(a^(2))/(p^(2))`