Question

1.) Eplain why, for a given object-screen distance, there are two positions where the

image is in focus.

Hints: (1) A quadratic equation is involved with the algebra and use the determinant

equation to verify the characteristics. There are three cases of where the determinant

equation is greater than, less than, and equal to zero.

Answer 1

The best explanation would be to draw the two situations.

The object can be further than (2 . f) from the lens when the image formed will be inverted, diminished and between f and 2f on the other side of the lens.

Or the object can be between f and 2f from the lens and the image will be inverted, magnified and beyond 2f on the other side of the lens.

You can point out that for the distant object, the light rays arriving at the lens are almost parallel ( having traveled a long distance, they must be nearly parallel - if they weren't they would have diverged enough so that they would miss the lens.) and so are strongly bent together and form an image not far from the lens and therefore not large.
The light from an object close to the lens (but not too close ) will still be diverging when it reaches the lens. The lens will bend the light back towards the principle axis but the image is now formed a long way beyond the lens and will consequently be enlarged.