Question

Let `alpha,beta` be the roots of the equation `ax^2+2bx+c=0` and `gamma,delta` be the roots of the equation `px^2+2qx+r=0` If `alpha,beta,gamma,delta` are in GP then `(a) q^2ac=b^2pr` `(b) qac=bpr` `(c) c^2pq=r^2ab``(d) p^2ab=a^2qp`

Answer 1

` alpha+beta=(-2b)/a`and `gamma+delta=(-2q)/p`
also `alphabeta=c/a and gammadelta=r/p`
given `alpha,beta gamma,delta` are in G.P so let them be `x,xy,xy^2,xy^3`respectively with x first term and y as common ratio.
` alpha+beta=(-2b)/a=x(1+y)`...(1) and `gamma+delta=(-2q)/p=xy^2(1+y)`....(2)
dividing (1) by(2) we get
`y^2=(alpha+beta)/(gamma+delta)=(bp)/(qa)`....(3)
`alphabeta=c/a=x^2y`....(4) and `gammadelta=r/p=x^2y^5`...(5)
dividing (3)by(4)=`1/y^4=(cp)/(ra)`...(6)
by (6) and (3)we get=>`q^2ac=b^2pr`