The distance between the object and the screen is $100 \, cm$. $A$ lens produces an image on the screen when it is placed at either of two positions $40 \, cm$ apart. The power of the lens is (approximately): (in $, D$)

Two point sources $P$ and $Q$ are $24 \ cm$ apart. Where should a convex lens of focal length $9 \ cm$ be placed in between them so that the images of both sources are formed at the same place?

The unit of focal power of a lens is

Consider a thin lens placed between a source $(S)$ and an observer $(O)$ (See figure). Let the thickness of the lens vary as $w(b) = w_0 - \alpha b^2$, where $b$ is the vertical distance from the pole. $w_0$ and $\alpha$ are constants. Using Fermat's principle, i.e., the time of transit for a ray between the source and observer is an extremum, find the condition that all paraxial rays starting from the source will converge at a point $O$ on the axis. Find the focal length.
$(ii)$ $A$ gravitational lens may be assumed to have a varying width of the form $W = K_1 \log \left( \frac{K_2}{b} \right)$ (where $b_{\min} < b < b_{\max}$). Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius $\beta = \sqrt{\frac{(n - 1)K_1 u}{v(u + v)}}$.

The focal length of a thin lens made from a material of refractive index $1.5$ is $15 \ cm$. When it is placed in a liquid of refractive index $\frac{4}{3}$,its focal length will be $..........\ cm$.

The distance between an object and the screen is $80 \, cm$. $A$ lens produces an image on the screen when the lens is placed at either of the positions $20 \, cm$ apart. The focal length of the lens is ....... $cm$.