What is x^2+3y^2-x-1=0 ?

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

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x^2+3y^2-x-1=0

x^{2} + 3 y^{2} - x - 1 = 0

(h, k) = (\frac{1}{2}, 0), a = \frac{5}{2}, b = \frac{5}{2 3}

Solution steps

x^{2} + 3 y^{2} - x - 1 = 0

Rewrite

x^{2} + 3 y^{2} - x - 1 = 0

in the form of the standard ellipse equation

\frac{( x - \frac{1}{2} ) ^{2}}{( \frac{5}{2} ) ^{2}} + \frac{( y - 0 ) ^{2}}{( \frac{5}{2 3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{2}, 0), a = \frac{5}{2}, b = \frac{5}{2 3}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{2}, 0), a = \frac{5}{2}, b = \frac{5}{2 3}

\frac{( x - 2 ) ^{2}}{20} + \frac{y ^{2}}{36} = 1

6 x^{2} + 9 y^{2} - 18 y - 45 = 0

\frac{( X - 1 ) ^{2}}{4} + \frac{( Y + 2 ) ^{2}}{16} = 1

4 x^{2} + 25 y^{2} - 50 y = 75

\frac{( x + 2 ) ^{2}}{4} + \frac{( y - 3 ) ^{2}}{9} = 1

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