Two astronomical telescopes P and Q have objective lenses of the same…

The correct answer is Option (1) → gathers 16 times as much light and has a higher resolving power

The light gathering power of a telescope depends on the area of its objective, which grows as the square of the diameter. The resolving power, the ability to see two objects in nearly the same direction as separate, also depends on the diameter of the objective. The focal length decides neither of them.

The ratio of the areas of the two objectives is:

    \( \begin{aligned} \frac{A_P}{A_Q} &= \left(\frac{D_P}{D_Q}\right)^{2} \\ &= \left(\frac{20\,\mathrm{cm}}{5\,\mathrm{cm}}\right)^{2} \\ &= 16 \end{aligned} \)

Option (1) gathers 16 times as much light and has a higher resolving power — Correct. The objective of P has 16 times the area of that of Q, so it collects 16 times as much light. Its larger diameter also gives it a higher resolving power.

Option (2) gathers 4 times as much light and has a higher resolving power — Incorrect. 4 is the ratio of the diameters. The light collected follows the area, which is the square of this ratio.

Option (3) gathers 16 times as much light but has the same resolving power — Incorrect. Equal focal lengths do not make the resolving power equal. It is set by the diameter of the objective, and the objective of P is four times as wide.

Option (4) gathers 4 times as much light but has the same resolving power — Incorrect. The light collected depends on the area, which gives a factor of 16. The wider objective of P also resolves better.

NCERT (2026-27):
Telescope – “The main considerations with an astronomical telescope are its light gathering power and its resolution or resolving power. The former clearly depends on the area of the objective. With larger diameters, fainter objects can be observed. The resolving power, or the ability to observe two objects distinctly, which are in very nearly the same direction, also depends on the diameter of the objective. …”