Answer:
12.
Step-by-step explanation:
The given equation is
![x^2+10x+y^2-4y=71](https://tex.z-dn.net/?f=x%5E2%2B10x%2By%5E2-4y%3D71)
It can be written as
![(x^2+10x)+(y^2-4y)=71](https://tex.z-dn.net/?f=%28x%5E2%2B10x%29%2B%28y%5E2-4y%29%3D71)
![(x^2+10x+5^2)+(y^2-4y+2^2)=71+5^2+2^2](https://tex.z-dn.net/?f=%28x%5E2%2B10x%2B5%5E2%29%2B%28y%5E2-4y%2B2%5E2%29%3D71%2B5%5E2%2B2%5E2)
![(x+5)^2+(y-2)^2=71+25+4](https://tex.z-dn.net/?f=%28x%2B5%29%5E2%2B%28y-2%29%5E2%3D71%2B25%2B4)
...(i)
The standard form of a circle is
...(ii)
where, (h,k) is center and r is radius.
From (i) and (ii), we get
![h=-5,k=2,r=10](https://tex.z-dn.net/?f=h%3D-5%2Ck%3D2%2Cr%3D10)
The center of circle is (-5,2) and radius is 10. So, the maximum value of y is
![Max(y)=k+r=2+10=12](https://tex.z-dn.net/?f=Max%28y%29%3Dk%2Br%3D2%2B10%3D12)
Therefore, the maximum value of y is 12.