How do I find the center of an ellipse with the equation 9x^2+16y^2-18x+64y=71?

Oct 7, 2014

Perform completing the Square...

Group the variables first

$9 {X}^{2} - 18 X + 16 {Y}^{2} + 64 Y = 71$

Do some factoring

$9 \left({X}^{2} - 2 X\right) + 16 \left({Y}^{2} + 4 Y\right) = 71$

Now, add something X and Y perfect squares.

$9 \left({X}^{2} - 2 X + 1\right) + 16 \left({Y}^{2} + 4 Y + 4\right) = 71$

However, since we added something on the left side of the equation,
we need to add the same value on the right side to maintain the equation.

$9 \left({X}^{2} - 2 X + 1\right) + 16 \left({Y}^{2} + 4 Y + 4\right) = 71 + 9 \left(1\right) + 16 \left(4\right)$

$9 {\left(X - 1\right)}^{2} + 16 {\left(Y + 2\right)}^{2} = 144$

From here it's already obvious that the ellipse is centered at (1, -2)