Question #129235
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
1
Expert's answer
2020-08-17T19:18:25-0400

9x2+4y2+36x24y+36=09(x2+4x+4)+4(y26y+9)36=09(x+2)2+4(y3)2=36(y3)29+(x(2))24=1a2=9,b2=4,c2=a2b2a=3,b=2,c=5y0=3,x0=2M(2,3);thefociarethepoints:F1(2,35),F2(2,3+5)theverticesare:V1(6,2),V2(0,2)V3(3,0),V4(3,4);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);


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