Given,

Circle $x^2+y^2-2x+4y-20=0$

Now center will be $\left( 1, -2\right)$

So, the line $4x-3y-10=0$ passes through center

Then length of the intercept equal to diameter of circle

So, radius $r=\sqrt{g^2+f^2-c}$$=\sqrt{1+4+20}=5$

So, diameter $d=2r=2\times 5=10$