Question

For what values of $a$ do the graphs of the functions $y=2ax+1$ and $y=\left(a-6\right)x^2-2$ not intersect?

Answer 1

$=>y=2ax+1...........\left(1\right)$

$=>y=(a-6)x^2-2...............\left(2\right)$

Substituting values of $y$ from $\left(1\right)$ into $\left(2\right),$

$=>(a-6)x^2-2ax-3=0$ (quadratic equation)

if the curves dont intersect they dont have a solution, So, $D<0$

Now $D$$\equiv 4a^2+12(a-6)<0$

$=>a^2+3a-18<0$

$=>(a+6)(a-3)<0$

$=>\forall aϵ(-6,3)$ curves $\left(1\right)$ and $\left(2\right)$ do not intersect.