**A small plane mirror is placed 21 cm in front of a concave mirror of focal length 21 cm. An object is placed 42 cm in front of the concave mirror.**

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**the whole question is here:**

**A small plane mirror is placed 21 cm in front of a concave mirror of focal length 21 cm. An object is placed 42 cm in front of the concave mirror. If light from the concave miror strikes the plane mirror, where is the final image?**

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First let us determine where is the image located with respect to the concave mirror. For a spherical mirror we have the equation

`1/x_1 +1/x_2 =1/f` , with the following signs

`x_1` is positive for an object being to the right of the mirror

`x_2` is positive for the image being to the right of the mirror

`f` is positive for concave mirrors

With the data in text we have

`x_1 =2f`

`1/(2f) + 1/x_2 =1/f`

`1/x_2 =1/(2f)`

`x_2 =2f` which means the image is exactly at the object position.

Now we insert the plane mirror at the focal point of the concave mirror. The image produced by the concave mirror is the object for the plane mirror.

For the plane mirror the equation of conjugate points is

`x1' = x2'` with the signs

`x_1'` is positive if the object is to the left of the mirror

`x_2'` is positive if the image is to the right of the mirror

With the data in text we have

`x_2' (=x_1') =f`

Now we can compute the position of the final image with respect to the initial object

`d = x1' +x2' =2f =2*21 =42 cm`

**The final image is 42 cm to the right of the initial object, thus it is formed exactly at the position where the concave mirror is located.**

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