y2 = 4ax ..........(1)
The equation to any straight line is
y = mx + c ........(2)
The coordinates of the points common to the straight line and the parabola satisfy both equations (1) and (2), and are therefore found by solving them.
Substituting the value of y from (2) in (1), we have
This is a quadratic equation for x and therefore has two roots, real, coincident, or imaginary.
The straight line therefore meets the parabola in two points, real, coincident, or imaginary.
The roots of (3) are real or imaginary according as
is positive or negative, i.e. according as - amc + a2 is
positive or negative, i.e. according as mc is ≤ 4a
