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

Question

11813 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-10-07T17:17:33

Group the variables first

#9X^2 - 18X + 16Y^2 + 64Y = 71#

Do some factoring

#9(X^2 - 2X) + 16(Y^2 + 4Y) = 71#

Now, add something X and Y perfect squares.

#9(X^2 - 2X + 1) + 16(Y^2 + 4Y + 4) = 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(X^2 - 2X + 1) + 16(Y^2 + 4Y + 4) = 71 + 9(1) + 16(4)#

#9(X - 1)^2 + 16(Y + 2)^2 = 144#

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