Right triangle $ABC$ has $BC=10,AC=24,AB=26$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$; a circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?

1. Place $C(0,0), A(24,0), B(0,10), O(x,10), P(24,y)$.
2. $OA=OB$: $(x-24)^2+100=x^2\Rightarrow x=\frac{676}{48}=\frac{169}{12}$; $PA=PB$: $576+(y-10)^2=y^2\Rightarrow y=\frac{676}{20}=\frac{169}5$.
3. $OP=\sqrt{\left(\frac{119}{12}\right)^2+\left(\frac{119}5\right)^2}=\boxed{\dfrac{1547}{60}}$.