Intersection of a straight line and a parabola

Parabola equation y² = 4ax,
Straight line equation y = mx+c

At intersection point, both equations are satisfied

hence (mx+c)² = 4ax
=> m²x²+2x(mc-2a)+c² = 0

It is a quadratic equation. Solution gives intersection points

The intersection points are concident if
4(mc-2a)² - 4m²c²>0
=> a² - amc>0
=>a-mc>0
=>a>mc
=>a/m>c
=>c
The intersection points are real and distinct if
4(mc-2a)² - 4m²c²=0
=> 4(mc-2a)² = 4m²c²

The intersection points are imaginary if
4(mc-2a)² - 4m²c²<0