This is a problem which has been bugging me for long. We know that a relation is called an equation to a curve if every point on the curve satisfies the equation and every point which satisfy the equation lies on the curve.
But when we derive the equation of a curve we only prove that every point on the curve satisfy the equation and we omit the converse.
For example, let us say we want to find the equation of a line making a given angle with the positive direction of axis of x and with a given intercept on the axis of y.
As you can see the converse is omitted. Also I haven't found any text which satisfactorly proves the converse. I tried proving it myself and it uses classical geometry. So my question is the converse not really necessary or can you provide a text in which the converse is proven in every case.