# No Intersection between Line and Parabola

The line $y = mx + c$ does not intersects the parabola ${y^2} = 4ax$ if $a < mc$.
Consider the standard equation of parabola with vertex at origin $\left( {0,0} \right)$can be written as

Also equation of a line is represented by

To find the point of intersection of parabola (i) and the given line (ii), using the method of solving simultaneous equation we solve equation (i) and equation (ii), in which one equation is in quadratic and other is in linear form, so take value of $y$ from equation (ii) and putting this value in equation (i) i.e. equation of parabola becomes

Since equation (iii) is a quadratic equation in $x$, and we can solve this quadratic equation either by completing square method or using quadratic formula. If equation (iii) has imaginary roots, then the line (ii) will not intersect the parabola (i), it is clear from the given diagram.
Equation (iii) will have imaginary if

Which is the required condition line is not intersects the parabola.