Let ` alpha ,beta " be the roots of " ax^(2) +bx + c= 0, gamma, detla " be the roots of " px^(2) + qx r=0 =0 and D_(1) and D_(2) ` be their respective discrimiant , if ` alpha, beta, gamma, delta` are in A.P., then the ratio ` D_(1) : D_(2)` is equal to

Options

A. `(a^(2))/(b^(2))`
B. `(a^(2))/(p^(2))`
C. `(b^(2))/(q^(2))`
D. `(c^(2))/(r^(2))`

Answer

B

Explanation

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))`