What is x^2-2x+3y^2-2y-5=0 ?

The solution to x^2-2x+3y^2-2y-5=0 is Ellipse with (h,k)=(1, 1/3),a=sqrt(19/3),b=(sqrt(19))/3

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x^2-2x+3y^2-2y-5=0

x^{2} - 2 x + 3 y^{2} - 2 y - 5 = 0

(h, k) = (1, \frac{1}{3}), a = \frac{19}{3}, b = \frac{19}{3}

Solution steps

x^{2} - 2 x + 3 y^{2} - 2 y - 5 = 0

Rewrite

x^{2} - 2 x + 3 y^{2} - 2 y - 5 = 0

in the form of the standard ellipse equation

\frac{( x - 1 ) ^{2}}{( \frac{19}{3} ) ^{2}} + \frac{( y - \frac{1}{3} ) ^{2}}{( \frac{19}{3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (1, \frac{1}{3}), a = \frac{19}{3}, b = \frac{19}{3}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (1, \frac{1}{3}), a = \frac{19}{3}, b = \frac{19}{3}

\frac{( x - 5 ) ^{2}}{8 ^{2}} + \frac{( y + 2 ) ^{2}}{1 0 ^{2}} = 1

2 y - x^{2} - y^{2} \geq 0

25 x^{2} + 4 y^{2} - 8 x + 200 y + 304 = 0

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

\frac{x ^{2}}{25} + \frac{( y - 2 ) ^{2}}{4} = 1

What is x^2-2x+3y^2-2y-5=0 ?
The solution to x^2-2x+3y^2-2y-5=0 is Ellipse with (h,k)=(1, 1/3),a=sqrt(19/3),b=(sqrt(19))/3