A Diameter Circle and a Fixed Length

An ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) passes through the point $P\left(\dfrac{\sqrt 6}2,\dfrac 12\right)$ and has eccentricity $\dfrac{\sqrt 2}2$. Let $M(2,t)$ ($t>0$) be a moving point, and let $O$ be the origin.

1. Find the standard equation of the ellipse. 2. Find the equation of the circle that has $OM$ as a diameter and is cut by the line $3x-4y-5=0$ in a chord of length $2$. 3. Let $F$ be the right focus of the ellipse. The line through $F$ perpendicular to $OM$ meets the circle with diameter $OM$ at a point $N$. Prove that the length of the segment $ON$ is a constant, and find this constant.