An isosceles triangle is formed with a thin rod of length `l_(1)` and coefficient of linear expansion `alpha_(1)`, as the base and two thin rods each of length `l_(2)` and coefficient of linear expansion `alpha_(2)` as the two sides. The distance between the apex and the midpoint of the base remain unchanged as the temperature is varied. the ratio of lengths `(l_(1))/(l_(2))` is

Options

A. `2 sqrt ((alpha_(1))/(alpha_(2)))`
B. `sqrt(alpha_(2)/alpha_(1))`
C. `2 sqrt(alpha_(2)/alpha_(1))`
D. `1/2 sqrt(alpha_(2)/alpha_(1))`

Answer

C

Explanation

The distance between the apex and the midpoint of the base, using Pythagoras theorem `l=sqrt((l_2)^(2)-((l_(1))/(2))^(2))`
or, `l^(2) = (l_2)^(2) - ((l_1)/(2))^(2)`
(i)
Differentiating (i) w.r.t. temperature

`0=2l_(2) xx(dl_2)/(dT) -2((l_1)/(2)) * 1/2 xx (dl_(1))/(dT)`
`(l_1)/(2) xx l_(1) alpha_(1) = 2l_(2) xx l_(2)alpha_(2)`
`((l_1)/(l_2))^(2) = 4 (alpha_2)/(alpha_1) rArr (l_1)/(l_2) = 2sqrt((alpha_2)/(alpha_1))`.