Question

According to the new cartesian sign convention, which sign is taken as origin?

Answer 1

Hint
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration. Rules for sign convention :
(i) All distances should be measured from the pole.
(ii) The distances measured in the direction of incident light, to be taken positive and measured in the opposite direction of incident light to be taken negative.
(iii) Height of object and height of image are positive if measured upward from the axis and negative if measured downward.
(iv) For a concave mirror '$f$’ and ‘$R$’ are negative and for a convex mirror these are positive.

Complete step-by-step answer:
According to the new Cartesian sign convention, the refraction of light through the concave and convex lens,all measured distances are measured from the optical centre of the lens. Also, According to the new cartesian sign convention, the pole of the mirror is taken as the origin. All the measurements of object distance, image distance and the focal length are recorded from the pole of the mirror. The measurement taken to the right side of the pole is positive whereas that to the left of the pole of the mirror is negative.

Note
The first time when sign convention is applied to derive Lens Formula, it is to make sure the formula is as general as possible. Meaning you may start deriving the formula for Convex lens but since you have ensured correct sign convention is used, it turns out the formula is valid for Concave lens too.