Let the line $\frac{x}{a}+\frac{y}{b}=1$ meets the co-ordinate axes in $A$ and $B$

$A=(a, 0) ; B=(0, b)$

Let $C$ be the centre of the circle circumscribing $\triangle O A B$. $C=\left(\frac{a}{2}, \frac{b}{2}\right)$ (Since circumcentre is the midpoint of hypotenuse in a right angled triangle) Radius $r=\frac{\sqrt{a^{2}+b^{2}}}{2}$

OC is perpendicular to the tangent at O. since, slope of $O C$ is $\frac{b}{a},$ slope of tangent at $O$ is $-\frac{a}{b}$ Equation of tangent is $a x+b y=0$

Let $d_{1}$ and $d_{2}$ be distances from $A$ and $B$ to the tangent respectively. $\rightarrow d_{1}=\frac{a^{2}}{\sqrt{a^{2}+b^{2}}}$ and $d_{2}=\frac{b^{2}}{\sqrt{a^{2}+b^{2}}}$

$\rightarrow d_{1}+d_{2}=\sqrt{a^{2}+b^{2}}=2 r$

Hence, diameter of the circle is $d_{1}+d_{2}$