The length of metallic wire is l _ 1 when tension in it is T _ 1 ....

Assuming Hooke's law to be valid

$T ∝ \left(\Deltaℓ\right)$

$T = k\left(\Deltaℓ\right)$

Let,${ℓ}_{0} =$ natural length (original length) $⇒ T = k\left(ℓ - {ℓ}_{0}\right)$

so, ${T}_{1} = k\left({ℓ}_{1} - {ℓ}_{0}\right) \text{ and } {T}_{2} = k\left({ℓ}_{2} - {ℓ}_{0}\right)$

$⇒ \dfrac{{T}_{1}}{{T}_{2}} = \dfrac{{ℓ}_{1} - {ℓ}_{0}}{{ℓ}_{2} - {ℓ}_{0}}$

$⇒ {ℓ}_{0} = \dfrac{{T}_{2}{ℓ}_{1} - {T}_{1}{ℓ}_{2}}{{T}_{2} - {T}_{1}}$