Can you help understand this by proving it mathematically or with an explanation

Question

Can you help understand this by proving it mathematically or with an explanation of what is occuring, thank you so much. 29) A lens of focal length f is used to project an image of a solar eclipse on a flat sheet of paper. Which of the following image distances gives the clearest image in this case ?
a) f/2      b) f      c) 2f      d) more than 10f      e) less than f/10 30) In order to produce an image double the size of an object with a converging lens of focal length f, what object distance should be used ? a) f       b) f/2      c) 2f      d) 2f/3      e) 3f/2 Can you help understand this by proving it mathematically or with an explanation of what is occuring, thank you so much. 29) A lens of focal length f is used to project an image of a solar eclipse on a flat sheet of paper. Which of the following image distances gives the clearest image in this case ?
a) f/2      b) f      c) 2f      d) more than 10f      e) less than f/10 30) In order to produce an image double the size of an object with a converging lens of focal length f, what object distance should be used ? a) f       b) f/2      c) 2f      d) 2f/3      e) 3f/2

Explanation / Answer

29) Given object distance p = 8 From thin lens formula    (1/f) = (1/p ) + (1/q) here f is focal length, p is object distance , q is image distance .
Putting the given value in the equation    (1/f) = 1/8 + (1/q)    (1/f) = 0 + (1/q) image distance q = f _______________________________________________________________________________ 30) Given    magnification m = 2    m = - q / p 2 = - q / p , so q = - 2p from thin lens formula    (1/f) = (1/p) + (1/q)    (1/f) = (1/p) + (1/-2p)
   (1/f) = 1/2p
object distance p = f /2

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