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Question

If Four distinct points (2,3),(0,2),(4,5) and (0,t) are concyclic, then t3+17 is equal to

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Solution

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+5x19y+34=0
As this circle passes through the point (0,t)
t219t+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

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