Ellipse

An ellipse is given by the equation 9x² + 25y² − 54x − 100y − 44 = 0. Find the lengths of the major and minor axes, the eccentricity, and the coordinates of the centre of the ellipse.

Final Answer:

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

Step-by-step explanation:

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

$o_2=2\cdot b=2\cdot 3=6$

$x_0=3$

$y_0=2$

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

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