A tacheometer was set at a station and the readings were taken on two points, which were 100 m and 150 m apart and had staff intercepts of 3.98 m and 5.98 m, respectively. If an externally focusing tacheometer was used for the observation, then the values of focal length (f), stadia interval (i) and distance (d) between the objective and the vertical axis of the instrument could be _________. 

CONCEPT

The horizontal distance between the axis and the staff is given by the following equation

D = f₁ + d = Ks + C = (f/i) * s + (f+d)

Where

i =  interval between stadia hairs 

S = staff intercept

f = focal length of the objective,

D = horizontal distance of the staff from the vertical axis of the instruments.

K = Multypling constant = f/i

C = Additive cocnstant = f+ d

Given 

S₁ = 3.98 m , S₂ = 5.98 m

D₁ = 100 m, D₂ = 150 m

Using the above given data we will form 2 equations to calculate the value of K(multiplying constant) and C(additive constant)

D₁ = K.s₁ + C     =>  100 = K x 3.98 + C

D₂ = K.s₂ + C     =>  150 = K x 5.98 + C

Solving the above equation we can find out the values of K and C,

K = 25 m & C = 0.5 m

f/i = 25 & f + d = 0.5

As the equation is only 2 and variables are 3 we will have to go by OPTION APPROACH.

Only the values given in option D follow the equation, hence it is the right answer (f = 0.3 m, i = 12 mm, d = 20 cm)

K = 25 m → K = f/i = 0.3×1000/12 = 25

C = 0.5 m → C = f + d = (0.3×100)+ 20 = 50 cm = 0.5 m