An object is 25cm away from lens and 30cm from mirror. Its image is exactly at object. what is focal point?

Question

The lens is convex and mirror is behind lens. Distance between lens and mirror is 5cm.

Answer 1

♠ see picture! (add ache-tee-tee-pee colon // in front and remove spaces first!)    

i174.photo bucket.com/ albums/w119/ kaksi_guy/Ase.png

Incident beam *a* will be reflected and return as *a’*, while incident beam *b* will return the same track and should pass through focus of lens;

Beam *a’* is not interesting for us; look at beams *a* and *b*!  

♣ geometry: h/H = 5/25; or; h=H/5;

(25-f)/H = f/h; or; (25-f)/H = f/(H/5); or;

25-f=5f, hence f=(25/6)cm;

Answer 2

The setup implies a kind of symmetry. The mirror is 5 cm to the right of the lens (assuming the convention where the object is to the left). If you were to remove the mirror and add an identical lens 10 cm to the right of the 1st lens, you'd have the same refractive arrangement, and the image of the object would be 25 cm to the right of the 2nd lens.

The first image formed is at the mirror's location. This is reflected back through the lens, resulting in the inverse transformation and forming a second image identical to the object. Therefore, f = 1/(1/di + 1/do) = 1/(1/25 + 1/5) = 1/0.24 = 4.166667 cm. The second image is upright and at the object's location.