A beam of light converges at a point P Now a lens is placed in the path of the convergent beam $12\text{cm}$ from P. At what point does the beam converge if the lens is

A) a convex lens of focal length $20\text{cm}$?

B) a concave lens of focal length $16\text{cm}$?

A) Step 1: Apply lens formula.

$u=+12\text{cm}$
For converging lens $f=+20\text{cm}$
By lens formula
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
Putting values,
$\frac{1}{v}-\frac{1}{12}=\frac{1}{20}$
$v=7.5\text{cm}$
Here the object is virtual, and the image is real (object on right; virtual).
Final Answer: $f=+20\text{cm}$. Image is real and at $7.5\text{cm}$ from the lens on its right side.

B) Step 1: Apply lens formula.

$u=+12\text{cm}$
For diverging lens, $f=-16\text{cm}$
By lens formula,
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
Putting values,
$\frac{1}{v}-\frac{1}{12}=-\frac{1}{16}$
$\frac{1}{v}=\frac{1}{12}-\frac{1}{16}$
$v=48\text{cm}$
Here the object is virtual, and the image is real (object on right; virtual).
Final Answer: $f=–16\text{cm}$. Image is real and at $48\text{cm}$ from the lens on its right side.