# An ambigious optical system with ambigious solutions [JEE Adv - 2016]

#### BlazenHammer

Homework Statement
A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convax
spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 40
cm. The mirror is tilted such that the axis of the mirror is at an angle θ = 30º to the axis of the
lens, as shown in the figure.

If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in
cm) of the point (x, y) at which the image is formed are
(A) (0, 0) (B) (50 – 25$\sqrt{3}$,25) (C) (25,25$\sqrt{3}$) (D) (125 /3,25$\sqrt{3}$)
Homework Equations
The usual formula in Geometrical Optics.(Although I'm doubtful if they are applicable to this problem)

This problem was asked in one of the most prestigious exams in India: JEE Advanced, unfortunately it was considered to be ambiguous for the scope of given examination and hence no official answer is issued for it as such.(BONUS marks to all)

And so many solutions were posted on Internet which are conflicting in physical principle with each other.

(i) Some are able to arrive at the (supposedly) correct answer (C)(25, $25\sqrt{3}$) (during the time of their publication) by applying mirror formula and assuming that rotation just changes orientation of reflected ray and preserves length, so image formed(by spherical mirror) is such that it radius vector is 50cm from mirror vertex just oriented 60 degree axis and proceed to do little trigonometry to arrive at answer.

(ii)Some of the solution on internet(again during the time of their publication) , first calculate the image formed by lens and then go on to calculate the image projection on the (now tilted) axis of spherical mirror, using that projection on axis as image, they apply the mirror formula , and locate the corresponding image which comes out in terms of some nasty surds and then assume since the only option closest to their answer is (C) hence they option C is correct

(iii) Recent one assumes this was a wrong question(hence awarded BONUS) as it was out of scope of syllabus.

What is my take on it?

I think (i) and (ii) are wrong because the mirror formula they use is only valid for paraxial rays since the mirror is tilted by 30 degrees so do the angle subtended by rays which ought to be paraxial increases by massive 30 degrees and after a little cheating(manual raytracing) I found that indeed it subtends a significant angle

Where am I stuck ?

Maybe I'm wrong and (i) and (ii) are correct as I'm not the most brightest of the lot?

Even If everyone is incorrect here and its out of scope of syllabus I wanna know how to tackle this kind of optical system , just for the sake of curiosity

I have one more question:

Is the projection of image voodo on the titled axis of spherical mirror done by (ii) guys is correct to apply mirror formula ? Is it physcially correct ( I feel uneasy to accept it because I don't think I've understood it correctly)

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