What is x^{(2)}+4y^{(2)}-10x-40y+109=0 ?

The solution to x^{(2)}+4y^{(2)}-10x-40y+109=0 is Ellipse with (h,k)=(5,5),a=4,b=2

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x^{(2)}+4y^{(2)}-10x-40y+109=0

x^{(2)} + 4 y^{(2)} - 10 x - 40 y + 109 = 0

(h, k) = (5, 5), a = 4, b = 2

Solution steps

x^{2} + 4 y^{2} - 10 x - 40 y + 109 = 0

Rewrite

x^{2} + 4 y^{2} - 10 x - 40 y + 109 = 0

in the form of the standard ellipse equation

\frac{( x - 5 ) ^{2}}{4 ^{2}} + \frac{( y - 5 ) ^{2}}{2 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (5, 5), a = 4, b = 2

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (5, 5), a = 4, b = 2

22 x^{2} + 6 y^{2} - 66 x + 60 y + 49 = 0

x^{2} + y^{2} + 6 y < 0

4 x^{2} + 9 y^{2} + 12 x - 18 y - \frac{83}{4} = 0

9 x^{2} + 25 y^{2} - 18 x - 100 y - 3 = 0

(\frac{x ^{2}}{a ^{2}}) + (\frac{y ^{2}}{b ^{2}}) = 1

What is x^{(2)}+4y^{(2)}-10x-40y+109=0 ?
The solution to x^{(2)}+4y^{(2)}-10x-40y+109=0 is Ellipse with (h,k)=(5,5),a=4,b=2