Question 307600
Write in standard form:
{{{x^2+9y^2-4x+54y+49 = 0}}} Group like-terms together:
{{{(x^2-4x)+(9y^2+54y)+49 = 0}}} Subtract 49 from both sides.
{{{(x^2-4x)+(9y2+54y) = -49}}} Factor out a 9 from the second group.
{{{(x^2-4x)+9(y^2+6y) = -49}}} Now complete the squares in x and y.
{{{(x^2-4x+4)+9(y^2+6y+9) = -49+4+81}}} Factor the left side and simplify the right side.
{{{(x-2)^2+9(y+3)^2 = 36}}} Now divide through by 36.
{{{(x-2)^2/36 + (y+3)^2/4 = 1}}} Rewrite as:
{{{highlight((x-2)^2/6^2 + (y+3)^2/2^2 = 1)}}}
You may recognize this as the equation of an ellipse, in standard form, whose major axis is parallel to the x-axis, with its center at (2, -3), its semi-major axis = 6 and its semi-minor axis = 2.
The standard form of the equation of an ellipse with its major axis parallel to the x-axis, center at (h, k), and {{{a^2 > b^2}}} looks like:
{{{(x-h)^2/a^2+(y-k)^2/b^2 = 1}}}