An isosceles triangles 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. `(l_(1))/(l_(2))=2sqrt((alpha_(2))/(alpha_(1)))`
B. `(l_(1))/(l_(2))=2sqrt((alpha_(1))/(alpha_(2)))`
C. `(l_(1))/(l_(2))=sqrt((alpha_(1))/(alpha_(2)))`
D. `(l_(1))/(l_(2))=sqrt((alpha_(2))/(alpha_(1)))`

Explanation

According to condition of the problem height of the isosceles triangle ABC is unchanged. The dotted lines show configuration after a temperature rise. Increase in length of AC is A'N
Considering increase in angle `theta` to be very small.
`A'N="AA"'costheta`
where cos `theta=(l_(1))/(2l_(2))`
thus we have `l_(1)alpha_(2)triangleT=((1)/(2)l_(1)alpha_(1)triangleT)((l_(1))/(2l_(2)))`
Hence `(l_(1))/(l_(2))=2sqrt((alpha_(2))/(alpha_(1)))`