If `a,b,c` are in GP, show that the equations `ax^(2)+2bx+c=0` and `dx^(2)+2ex+f=0` have a common root if `a/d,b/e,c/f` are in

Options

A. AP
B. GP
C. HP
D. None of these

Answer

A

Explanation

Given that a,b,c are GP.
Then, `b^(2)=ac " " "…..(i)"` and equations `ax^(2)+2ex+c=0`
and `dx^(2)+2ex+f=0 " have a common root. ""………..(A)"`
Now, `ax^(2)+2bx+c=0`
`implies ax^(2)+2sqrt(ac)" " x+c=0 " " [" by Eq. (i) "]`
`implies (sqrt(ax)+sqrt(c ))^(2)=0 " " implies sqrt(ax)+sqrt(c )=0`
`implies x=-(sqrt(c ))/(sqrt(a)) " " [" repeated "]`
By the condition `(A), (-(sqrt(c ))/(sqrt(a)))` be the root of `dx^(2)-2ex+f=0`
So, it saftisfy the equation
`d(-sqrt(c/(a)))^(2)+2e(-sqrt(c/(a)))+f=0`
` implies (dc)/(a)-2esqrt(c/(a))+f=0 " " implies (d)/(a)-(2e)/(sqrt(ac))+(f)/(c )=0`
` implies (d)/(a)-(2e)/(b)+(f)/(c )=0 " " implies (d)/(a)+(f)/(c )=2((e)/(b))`
So, `(d)/(a),(e)/(b),(f)/(c )` are in AP.