Two antennas A and B have maximum line of sight range R_AB = 45.5 km…

The correct answer is Option (3) → 50 m

An antenna of height h can send a signal in a straight line (line of sight) only up to the point where the ray just grazes the earth. If R is the radius of the earth, this gives \( d^{2} = (R+h)^{2} - R^{2} = 2Rh + h^{2}\approx 2Rh \), since \( h\ll R \). For two antennas the maximum range is the sum of their two distances.

    \( R_{AB} = \sqrt{2Rh_A} + \sqrt{2Rh_B} \)

    \( R = 6400\,\mathrm{km} = 6.4\times10^{6}\,\mathrm{m},\quad R_{AB} = 45.5\,\mathrm{km} = 45500\,\mathrm{m} \)

    \( \sqrt{2Rh_A} = \sqrt{2\times6.4\times10^{6}\times32}\,\mathrm{m} = \sqrt{4.096\times10^{8}}\,\mathrm{m}\approx 20240\,\mathrm{m} \)

    \( \sqrt{2Rh_B} = (45500 - 20240)\,\mathrm{m} = 25260\,\mathrm{m} \)

    \( \displaystyle h_B = \frac{(25260)^{2}}{2\times6.4\times10^{6}}\,\mathrm{m} = \frac{6.38\times10^{8}}{1.28\times10^{7}}\,\mathrm{m}\approx 50\,\mathrm{m} \)

Shortcut: Put the options back in: for \( h_B = 50\,\mathrm{m} \), \( \sqrt{2R}\,(\sqrt{32} + \sqrt{50}) = 3578\times(5.66 + 7.07)\,\mathrm{m}\approx 45.5\,\mathrm{km} \), which is the given range, so Option (3) fits at once.

Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.

NCERT (Pre-rationalised edition, 2022-23):
Space Wave – “… If the transmitting antenna is at a height h_T, then you can show that the distance to the horizon d_T is given as \( d_T = \sqrt{2Rh_T} \), where R is the radius of the earth (approximately 6400 km). … With reference to Fig. 15.5 the maximum line-of-sight distance d_M between the two antennas having heights h_T and h_R above the earth is given by

    \( \displaystyle d_M = \sqrt{2Rh_T} + \sqrt{2Rh_R} \qquad (15.1) \)

where h_R is the height of receiving antenna.”