Question

Use the mirror equation to show that a convex lens always produces a virtual image independent of the location of the object.

Answer 1

For a convex  mirror, focal length(f) is always positive  and object distance (u) is always negative according to the convention.
We have to show that the image formed is virtual. A virtual image means image is formed behind the mirror or image distance(v) is always positive.

Mirror formula is given as
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Thus we can write
$\frac{1}{v}= \frac{1}{f}- \frac{1}{u}$

here, $\frac{1}{f}$ is positive(since f is positive) and $-\frac{1}{u}$ is also positive(as u is negative). Since $\frac{1}{v}$ is sum of two positive quantities, $\frac{1}{v}$always positive. So v, the image distance is always positive or the image is always virtual.