had to add an extra condition to get as many equations as variables
In[13]:= sln =
NDSolveValue[{(x[t] - Sin[t])/(y[t] - Exp[t]) == y'[t]/x'[t],
x'[t]^2 + y'[t]^2 == 1, x[0] == 1, y[0] == 2}, {x[t], y[t]}, {t, 0,
2}]
During evaluation of In[13]:= NDSolveValue::ndsvb: There are multiple solution branches for the equations, but NDSolveValue will return only one. Use NDSolve to get all of the solution branches.
ParametricPlot[sln, {t, 0, 2}]