I\'m having trouble with the following question: A bright object is placed on on

Question

I'm having trouble with the following question:
A bright object is placed on one side of a converging lens offocal length f, and a white screen for viewing the image is on theopposite side. The distance Dt = Di + Do between the objectand the screen is kept fixed, but the lens can be moved (a) Show that if Dt is greater than 4f, there will be twopositions where the lens can be placed and a sharp image will beproduced on the screen. (b) If Dt is less than 4f, show that there will be no lensposition where a sharp image is formed (c) Determine a formula for the distance between the two lenspositions in part (a), and the ratio of the image sizes
Any help would be greatly appreciated!
A bright object is placed on one side of a converging lens offocal length f, and a white screen for viewing the image is on theopposite side. The distance Dt = Di + Do between the objectand the screen is kept fixed, but the lens can be moved (a) Show that if Dt is greater than 4f, there will be twopositions where the lens can be placed and a sharp image will beproduced on the screen. (b) If Dt is less than 4f, show that there will be no lensposition where a sharp image is formed (c) Determine a formula for the distance between the two lenspositions in part (a), and the ratio of the image sizes
Any help would be greatly appreciated!

Explanation / Answer

(a)    the thin lens equation with the distances isgiven by    do + di =dT    (1 / do) + (1 / di) = (1 /f)    (1 / do) + (1 / (dT -do)) = (1 / f)    when we solve we get a quadratic equation indo    do2 - dTdo + dT f = 0    do = (1 / 2) [dT ±(dT2 - 4 dTf)1/2]    if dT > 4 f then the term insidethe square root dT2 - 4 dT f >0 and    (dT2 - 4 dTf)1/2 < dT (b)    if dT < 4 f then the terminside the square root dT2 - 4 dTf < 0    as a result there are no real solutions fordo (c)    when there are two solutions thedistance betweenthem will be    d = do1 -do2            (c)    when there are two solutions thedistance betweenthem will be    d = do1 -do2           

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