1.

If p, q, r are in G.P. and the equations, `p x^2+2q x+r=0`and `dx^2+2e x+f=0`have a common root, then show that `d/p`,`e/q`,`f/r`are in A.P.

Answer» Since, p, q, r are in GP, we have
`q^(2)=pr` ...(i)
On solving `px^(2)+2qx+r=0`, we get
`x=(-2q pm sqrt(4q^(2)-4pr))/(2p)=(-2q)/(2p)=(-q)/p` [using (i)]
Thus, `x=(-q)/p` is a repeated root of `px^(2)+2pr+r=0`.
`:. x=(-q)/p` is also a root of `dx^(2)+2ex+f=0`
`rArr d.((-q)/p)^(2)+2e((-q)/p)+f=0`
`rArr dq^(2)-2eqp+fp^(2)=0` ...(ii)
`rArr d/p-(2e)/q+(fp)/q^(2)=0` [on dividing (ii) by `pq^(2)`]
`rArr d/p+f/r=(2e)/q`.
Hence, `d/p, e/q, f/r` are in AP.


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