Let equation of the circle be x2+y2+2gx+2fy+c=0 .....(1)
Since it passes through the points (2,3),(0,2),(4,5)
We have 4g+6f+c=−13 ....(2)
4f+c=−4 ....(3)
8g+10f+c=−41 ....(4)
Solving these equations, we get
g=52,f=−192,c=34
So, the equation of the circle is x2+y2+5x−19y+34=0
As this circle passes through the point (0,t)
t2−19t+34=0
⇒t=2 or t=17.
But t=2 corresponds to the point (0,2) which is different from (0,t) Therefore t=17
So, t3+17=4930