If the parabolas `y^2=4a x` and `y^2=4c(x-b)` have a common normal other than the x-axis `(a , b , c` being distinct positive real numbers), then prove that `b/(a-c)> 2.`

Explanation

which parabola `y^(2)=4ax`, equation of normal having slope m is
`y=mx-2am-am^(3)` (1)
For parabola `y^(2)=4c(x-b)`, equation of normal having slope m is `y=m(x-b)-2cm-cm^(3)` (2)
For common normal to parabolas, equations (1) and (2) are identical.
`:." "-2am-am^(3)=-bm-2cm-cm^(3)`
`rArr" "(c-a)m^(2)=2(a-c)-b" (as m" !=0)`
`rArr" "m^(2)=(2(a-c)-b)/(c-a)`
For real value of m,
`(2(a-c)-b)/(c-a)gt0`
`rArr" "-2+(b)/(a-c)gt0`
`rArr" "(b)/(a-c)gt2`