What is x^2+y^2+10y=0 ?

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

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x^2+y^2+10y=0

x^{2} + y^{2} + 10 y = 0

(h, k) = (0, - 5), a = 5, b = 5

Solution steps

x^{2} + y^{2} + 10 y = 0

Rewrite

x^{2} + y^{2} + 10 y = 0

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{5 ^{2}} + \frac{( y - ( - 5 ) ) ^{2}}{5 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, - 5), a = 5, b = 5

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (0, - 5), b = 5, a = 5

12 x^{2} + 96 x + 5 y^{2} + 50 y + 257 = 0

\frac{( x - 3 ) ^{2}}{4} + \frac{( y + 2 ) ^{2}}{3} = 1

\frac{( x - 2 ) ^{2}}{81} + \frac{( y + 5 ) ^{2}}{64} = 1

\frac{( x + 2 ) ^{2}}{9} + \frac{( y - 3 ) ^{2}}{25} = 1

\frac{x ^{2}}{1} + \frac{y ^{2}}{2} = 1

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