If I have a linear function and some kind of quadratic in x and y ie: $x^2+xy+y^2=1$ that share two roots, then I can substitute that linear function into the quadratic expression and use the Sum of Roots formulae to find the midpoint of the chord.
My question is: Is there a similar geometric application of the Product of Roots?
If you want to find the difference between the two intersections, then you may want to use the formula
$$ |x_1 - x_2| = \sqrt{(x_1 + x_2)^2 - 4x_1x_2}$$