Defining Conic Section From 5 Points (No Matrix work)

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I am doing a homework assignment, and I need to find:

a) what type of conic it is b) the equation for the conic

from $5$ points. In class we haven't done any work with matrices, so I need a method which does not use them.

Example problem: $(0,0), (1,5), (3,4), (2,0), (2,3)$

Any help is greatly appreciated. Thanks.