Q: Find the equation of the parabola if its tangent line at x=2 is
$\displaystyle y=\frac{2}{\sqrt{3}}x-\frac{\sqrt{3}}{3}$
TYIA
At x= 2 y= $\displaystyle \sqrt{3}$ so you are asking for a parabola that passes through $\displaystyle (2,\sqrt{3})$ and has derivative $\displaystyle \frac{2}{\sqrt{3}}$ there. As skeeter says, there are an infinite number of such parabolas. Even if we restrict to parabolas with vertical axis, there are an infinite number of answers.
$\displaystyle \sqrt{x^2-1} = ax^2+bx+c$ at $\displaystyle x = 2$
derivatives ...
$\displaystyle \frac{x}{\sqrt{x^2-1}} = 2ax + b$
... are also equal at $\displaystyle x = 2$.
since the question said "a" parabola, let $\displaystyle c = 0$.
get two equations in terms of $\displaystyle a$ and $\displaystyle b$ and solve the system for the coefficients.