Statement –1 : The only circle having radius $\sqrt{10}$ and a diameter along line $2x+y=5$ is $x^2+y^2-6x+2y=0.$

Statement-2: $2x+y=5$ is a normal  to the circle $x^2+y^2-6x+2y=0$

Circle: $x^2+y^2-6x+2y=0$

Line: $2x+y=5$

Centre $=(3,-1)$

Now, $2\times 3-1=5,$ hence centre lies on the given line.  

Therefore line passes through the centre. The given line is normal to the circle. This statement-2 is true, but statement-1 is not true as there are infinite circle according to the given conditions.