# 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.

Result

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

#### Solution:

$(\dfrac{ x-x_{ 0 } }{ a } )^2+(\dfrac{ y-y_{ 0 } }{ b } )^2 = 1 \ \\ (9x^2-54x)+(25y^2-100y) = 44 \ \\ 9 \cdot \ (x^2-6x)+25(y^2-4y) = 44 \ \\ \ \\ x^2-6x+9 = (x-3)^2 \ \\ y^2-4y+4 = (y-2)^2 \ \\ \ \\ 9 \cdot \ (x^2-6x+9)+25(y^2-4y+4) = 44+9 \cdot \ 9+4 \cdot \ 25 \ \\ \ \\ 9 \cdot \ (x-3)^2+25 \cdot \ (y-2)^2 = 225 \ \\ \dfrac{ 9 }{ 225 } \cdot \ (x-3)^2+\dfrac{ 25 }{ 225 } \cdot \ (y-2)^2 = 1 \ \\ \ \\ a = \sqrt{ \dfrac{ 225 }{ 9 } } = 5 \ \\ b = \sqrt{ \dfrac{ 225 }{ 25 } } = 3 \ \\ \ \\ o_{ 1 } = 2 \cdot \ a = 2 \cdot \ 5 = 10$
$o_{ 2 } = 2 \cdot \ b = 2 \cdot \ 3 = 6$
$x_{ 0 } = 3$
$y_{ 0 } = 2$
$e = \sqrt{ a^2-b^2 } = \sqrt{ 5^2-3^2 } = 4$

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Tips to related online calculators
For Basic calculations in analytic geometry is helpful line slope calculator. From coordinates of two points in the plane it calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of segment, intersections the coordinate axes etc.
Pythagorean theorem is the base for the right triangle calculator.

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