Question

let alpha and beta be the roots of ax²+bx+c=0 and gamma and delta be the roots of px² + qx + r= 0. if alpha, beta, gamma, delta are in hp then:
(A) (b²- 4ac)/(q²- 4pr) = c²/r²
(B) α-β/αβ = 1/4(b/c - q/r)
(C) gamma- delta/gamma* delta = 1/4(b/c - q/r)
(D) (b² - 4ac)/(q² - 4pr) = a²/p²

Answer 1

Answer: (A)If α ,β and ¥,$are the roots of first and second equation respectively ,then

α + β = -b/a ,α.β = c/a

¥ + $ = -q/p , ¥.$ = r/p

(α- β)² = ( α + β )² - 4αβ

= (- b/a )² - 4c/a

= b² - 4ac / a² ....... (1)

Similarily ,

( ¥- $ )² = q²- 4pr / p² ....(2)

Divide 1st and 2nd euation:

b²- 4ac / q²- 4pr = a²/p² * ( α-β /¥-$)² .(3)

Now ,

If α ,β,¥ and $ are in H.P. (i.e. reciprocal of the α,β,¥,$ are in A.P )

i.e. 1/ α , 1/β ,1/¥ ,1/$ are in A.P.

Since Common difference in A.P is same ...So.

1/β - 1/α = 1/¥ - 1/β = 1/$ - 1/¥

euating and solving here first and third terms , we get ,

α-β/ αβ = ¥-$/¥$

=> α- β / ¥-$ = αβ/ ¥$

=> α-β/¥-$ = pc / ar ......(4)

Now , put the above value of the α-β/¥-$ in 3rd euation...we get ,

b²-4ac / q²-4pr = c²/ r²

Answer 2

hope it help uh↑↑↑↑↑................