Let a be a non-zero real number and `alpha, beta` be the roots of the equation `ax^(2) + 5x + 2 = 0`. Then the absolute value of the difference of the roots of the equation `a^(3)(x+5)^(2) - 25 a(x+5) + 50 = 0`, is

Options

A. `|alpha^(2) - beta^(2)|`
B. `|alpha beta (alpha^(2) - beta^(2))|`
C. `|(alpha^(2) - beta^(2))/(alpha beta)|`
D. `|(alpha^(2) - beta^(2))/(alpha^(2) beta^(2))|`

Answer

A

Explanation

Clearly, `alpha + beta = -(5)/(a) and alpha beta = (2)/(a)`
Let p, q be the roots of the equation `a^(3) x^(2) + 5a (2a^(2)-5) x + 25(a^(3) - 5a + 2) = 0`.
Then, `p+q = -(5)/(a^(2))(2a^(2) - 5) and, pq = (25)/(a^(3))(a^(3) - 5a + 2)`
`therefore" "|p-q| = sqrt((p+q)^(2)-4pq)`
`rArr" "|p-q|=sqrt((25)/(a^(4))(2a^(2)-5)^(2)-(100)/(a^(3))(a^(3)-5a + 2))`
`rArr" "|p-q|=(5)/(a^(2))sqrt(4a^(2)-20a^(2)+25-4a^(4)+20a^(2)-8a)`
`rArr" "|p-q|=(5)/(a^(2))sqrt(25-8a)=sqrt(((5)/(a))^(4)-4((5)/(a))^(2)xx(2)/(a))`
`rArr" "|p-q|=sqrt((alpha+beta)^(4)-4(alpha+beta)^(2)alpha beta)`
`rArr" "|p-q|=|alpha+beta|sqrt((alpha+beta)^(2)-4alpha beta)`
`rArr" "|p-q|=|alpha+beta||alpha-beta|=|alpha^(2)-beta^(2)|`