Parabola Properties - Straight line & Intercept

Take a line between any two points (a, a²) and (b, b²) on the parabola y=x² What the connection between the two points and the y intercept of the line? Can you prove your result algebraically in terms of a & b? This is one of the many suprisingly neat properties to be found in the geometry of the simple parabola.
Hint: Using (a, a²) and (b, b²) you might want to consider the gradient and the equation of the line between the two points. Not the only way of course, as an alternative you could consider similar triangles and ratios.