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We use collective oscillations of a two-component Bose-Einstein condensate(2CBEC) of \Rb atoms prepared in the internal states $\ket{1}\equiv\ket{F=1,m_F=-1}$ and $\ket{2}\equiv\ket{F=2, m_F=1}$ for the precision measurement ofthe interspecies scattering length $a_{12}$ with a relative uncertainty of$1.6\times 10^{-4}$. We show that in a cigar-shaped trap the three-dimensional(3D) dynamics of a component with a small relative population can beconveniently described by a one-dimensional (1D) Schr\"{o}dinger equation foran effective harmonic oscillator. The frequency of the collective oscillationsis defined by the axial trap frequency and the ratio $a_{12}/a_{11}$, where$a_{11}$ is the intra-species scattering length of a highly populated component1, and is largely decoupled from the scattering length $a_{22}$, the total atomnumber and loss terms. By fitting numerical simulations of the coupledGross-Pitaevskii equations to the recorded temporal evolution of the axialwidth we obtain the value $a_{12}=98.006(16)\,a_0$, where $a_0$ is the Bohrradius. Our reported value is in a reasonable agreement with the theoreticalprediction $a_{12}=98.13(10)\,a_0$ but deviates significantly from thepreviously measured value $a_{12}=97.66\,a_0$ \cite{Mertes07} which is commonlyused in the characterisation of spin dynamics in degenerate \Rb atoms. UsingRamsey interferometry of the 2CBEC we measure the scattering length$a_{22}=95.44(7)\,a_0$ which also deviates from the previously reported value$a_{22}=95.0\,a_0$ \cite{Mertes07}. We characterise two-body losses for thecomponent 2 and obtain the loss coefficients${\gamma_{12}=1.51(18)\times10^{-14} \textrm{cm}^3/\textrm{s}}$ and${\gamma_{22}=8.1(3)\times10^{-14} \textrm{cm}^3/\textrm{s}}$.

Precision measurements of s-wave scattering lengths in a two-component Bose-Einstein condensate