`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.`
If the roots of the equation `x^(2) + px+ q = 0` are in the same ratio as those of the equation `x^(2) +lx + m = 0`, then which one of the following is correct ?

Options

A. `p^(2) m = l^(2) q`
B. `m^(2)p = l^(2) p`
C. ` m^(2) p = q^(2) l `
D. ` m^(2)p^(2) = l^(2) q`

Answer

A

Explanation

Let a,b be the roots of
` x^(2) + p//x^(2) + q = 0 `
So, ` a+b = - p//x^(2) , ab=q" "` …(i)
Let c,d be the roots of
`x^(2) + lx + m = 0 `
So, `c+d = - 1, cd = m`
Given that roots of both the equations are n the Same ratio.
So, `a/b = c/d " "` ...(iii)
`rArr b/a = d/c " "` ...(iv)
`(iii) + (iv) rArr a/b +b/c = c/d + d/c `
` rArr (a^(2) + b^(2)) /(ab) = (c^(2) + d^(2) + 2cd)/(cd)`
`rArr (a+b)^(2) /ab = (c+d)^(2) /(cd)`
`rArr (-P)^(2)/q = (-1)^(2)/m" " `(from (i) and (ii))
`rArr P^(2)m = 1^(2)q.`