simplify to get 16 * (x+2)^2 - 64 + 9 * (y-3)^2 - 81 = -1
add 64 and 81 to both sides of the equation to get:
16 * (x+2)^2 + 9 * (y-3)^2 = 144
divide both sides of the equation by 144 to get:
16 * (x+2)^2 / 144 + 9 * (y-3)^2 / 144 = 1
this can be written as 16/144 * (x+2)^2 + 9/144 * (y-3)^2 = 1
simplify so that the numerator in the fractions is 1 to get (x+2)^2 / 9 + (y-3)^2 / 16 = 1
this is now in standard form of (x-h)^2 / b^2 + (y-k)^2 / a^2 = 1
the designation of the letter a always goes where the largest denominator is and the designation of the letter b always goes where the smallest denominator is.
that's why the b^2 went under the (x-h)^2 and the a^2 went under the (y-k)^2.
in your equation, you have:
a^2 = 16 which makes a = 4
b^2 = 9 which makes b = 3
center of the ellipse is (h,k) which make the center (-2,3).
the major axis of the ellipse is the axis that is the longest.
a is the distance along the major axis from the center of the ellipse to the vertex of the ellipse.
b is the distance along the minor axis from the center of the ellipse to the co-vertex of the ellipse.
c is the distance along the major axis from the center of the ellipse to the the focus of the ellipse.
c is calculated using the formula c^2 = a^2 - b^2.
in your equation, c^2 is therefore equal to 9 - 4 = 5.
this makes c = sqrt(5).
your important values to the ellipse are now:
a = 4
b = 3
c = sqrt(5)
center = (-2,3)
the graph of your ellipse looks like this:
here's a reference that should help you understand what's going on.
