Answer to Question #129235 in Analytic Geometry for Sym

Question

Given the ellipse whose equation is
9x² +4y² + 36x -24y+36=0 find
The coordinates of its centre
The length of the major and minor axes
The coordinates of its vertices and foci

Answer 1

$9x^2+4y^2+36x-24y+36=0\\ 9(x^2+4x+4)+4(y^2-6y+9)-36=0\\ 9(x+2)^2+4(y-3)^2=36\\ \frac{(y-3)^2}{9}+\frac{(x-(-2))^2}{4}=1\\ a^2=9,\quad b^2=4,\quad c^2=a^2-b^2\\ a=3,\quad b=2,\quad c=\sqrt{5}\\ y_0=3,\quad x_0=-2\Rightarrow M(-2,3);\\ the \quad foci \quad are \quad the \quad points: \quad F_1(-2,3-\sqrt{5}),\quad F_2 (-2,3+\sqrt{5})\\ the \quad vertices \quad are :\quad V_1(6,-2),\quad V_2(0,-2)\\ V_3(3,0),\quad V_4(3,-4);$