Given the ellipse 8x^2 + y^2 + 80x - 6y + 193 = 0 find:

a) The Center C

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To find the center of an ellipse, it needs to be put into standard form. This means that you need complete the square for both the x and y coordinates.

`8x^2+y^2+80x-6y+193=0` collect terms for x and y

`8x^2+80x+y^2-6y+193=0` factor 8 from x terms

`8(x^2+10x)+y^2-6y+193=0` add and subtract to complete the square

`8(x^2+10x+25-25)+y^2-6y+9-9+193=0` arrange perfect squares

`8(x+5)^2-8(25)+(y-3)^2-9+193=0` collect constants

`8(x+5)^2+(y-3)^2=200+9-193` simplify

`8(x+5)^2+(y-3)^2=16` divide by 16

`(x+5)^2/2+(y-3)^2/16=1`

**The centre of the ellipse is at (-5,3).**

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