Where does the discriminant come from? It is from the quadratic formula. It is the part under the square root sign, in other words it can't be negative or there are no real solutions to your quadratic equation.We also know that if it equals 0 there is 1 repeated solution, and if it is greater than 0 then there are 2 distinct solutions.So how does this all relate to coordinate geometry?Well, remember how you can talk about solving equations as the intersection of curves/lines? For example solving 2x = 3x+ 4 could be interpreted as finding the intersection of the line y = 2x and the line y = 3x + 4.So now say you had a circle and a line. When you solve these simultaneously algebraically, you are really finding where the graphs intersect!So, if you were to solve them simultaneously you would end up with a quadratic (either in terms of x or y depending on what you substitute) - but hang-on. What does it mean then, if your quadratic that you end up with has 2 solutions? or 1 solution? or no solutions?Well, it would mean that the line goes straight through the circle, or just touches it (is a tangent), or doesn't intersect it at all.