Ellipse

Ellipse is expressed by equation 9x² + 25y² - 54x - 100y - 44 = 0. Find the length of primary and secondary axes, eccentricity, and coordinates of the center of the ellipse.

Correct result:

o₁ =  10
o₂ =  6
x₀ =  3
y₀ =  2
e =  4

Solution:

$$\begin{gathered} ({\displaystyle \frac{x-x_0}{a}}{)}^2+({\displaystyle \frac{y-y_0}{b}}{)}^2=1 \\ (9x^2-54x)+(25y^2-100y)=44 \\ 9\cdot (x^2-6x)+25(y^2-4y)=44 \\ {x}^2-6x+9=(x-3{)}^2 \\ {y}^2-4y+4=(y-2{)}^2 \\ 9\cdot (x^2-6x+9)+25(y^2-4y+4)=44+9\cdot 9+4\cdot 25 \\ 9\cdot (x-3{)}^2+25\cdot (y-2{)}^2=225 \\ {\displaystyle \frac{9}{225}}\cdot (x-3{)}^2+{\displaystyle \frac{25}{225}}\cdot (y-2{)}^2=1 \\ a=\sqrt{{\displaystyle \frac{225}{9}}}=5 \\ b=\sqrt{{\displaystyle \frac{225}{25}}}=3 \\ {o}_1=2\cdot a=2\cdot 5=10 \end{gathered}$$

$o_{2}=2 \cdot \ b=2 \cdot \ 3=6$

$x_{0}=3$

$y_{0}=2$

$e=\sqrt{ a^2-b^2 }=\sqrt{ 5^2-3^2 }=4$

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