I am reading the book Complex algebraic curves by Kirwan. In the exercise of chapter 2 of the book i am asked to find the tangent line at a singular point.
I was able to calculate the singular point. I want a definition of tangent line, so that i can calculate the same? what is the geometric meaning of a tangent line here? Is it related in any way to theory of smooth manifolds?
Can someone please explain with minimal use of new math. Thank you!
$$x^4+y^4-x^2y^2=(x^2+y^2)^2-3x^2y^2=(x^2+y^2+\sqrt3xy)(x^2+y^2-\sqrt3xy)$$ which you can further factor as four lines, which are their own tangents.