If the length of radius of a right circular cylinder is doubled and height is halved, the lateral surface area will be

The lateral (or curved) surface area of a right circular cylinder with radius $r$ and height $h$ is given by:
$$\text{LSA}=2\pi rh$$

Let the new radius be $r^{\prime }=2r$ and the new height be $h^{\prime }=\frac{h}{2}$.
The new lateral surface area ${\text{LSA}}^{\prime }$ is:

$${\text{LSA}}^{\prime }=2\pi r^{\prime }{h}^{\prime }=2\pi (2r)\left(\frac{h}{2}\right)=2\pi rh=\text{LSA}$$

Thus, the lateral surface area remains equal.