My answer to a question on Quora: Why do I need to learn conics in math?
This is the second best kept secret of mathematics education: it does not matter what students are taught, what matters is how deep their learning, and how efficient is the network of connections between mathematical facts that grows in their minds in their minds: in how many steps they can get form Fact A to Fact B? Conics could be included in high school courses of mathematics, and could be omitted, it does not really matter. What matters is whether students develop specific mental skills of mathematical thinking.
In one of the best mathematics high schools in the world, conics are used as a training ground of mathematics problem solving, that is, solving problems not seen by students ever before — simply because the theory of conics, if taken seriously, is rich, and because it is really hard to find in the literature good books on advanced level but elementary theory of conics. Students, I was told, really have to work from scratch.
I can offer you one problem; I think it was used at one of the International Mathematical Olympiads of yesteryear.
Assume you are given a sheet of paper with a parabola printed on it. Using only straightedge and compasses, construct the Cartesian coordinate system in which this parabola has equation y=x^2.
People who can solve this problem have reasonably deep understanding of conics.