Let `a, x, b` in A.P, `a, y, b` in GP `a, z, b` in HP where `a` and `b` are disnict positive real numbers. If `x = y + 2` and `a = 5z`, then which of the following is/are `COR RECT` ?

Options

A. `y^(2) = zx`
B. `x gt y gt z`
C. `a = 9 , b = 1`
D. `a = (1)/(4), b = -(9)/(4)`

Answer

A, B, C

Explanation

`{:(a_(1)x_(1)b,"inA.P.",rArr,2x=a+b),(a_(1)y_(1)b,"inG.P.",rArr,y^(2)=ab),(a_(1)z_(1)b,"inA.P.",,):}`
`rArr z = (2ab)/(a + b)`
`:. x gt y gt z (A gt G gt H)`
at `y^(2) = xz`
`:. a = 5z` , (Given)…….(i)
`(a)/(5) = (2ab)/(a + b)` or `a = 9b`
`x = y + 2`
or `(a + b)/(2) = sqrt(ab) + 2`
or `5b = 3b + 2 , (b gt 0)`
`:. (1,2,3)`