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

The solution to 3x^2+y^2-5x=2-y is Ellipse with (h,k)=(5/6 ,-1/2),a=(sqrt(13))/3 ,b=sqrt(13/3)

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

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

(h, k) = (\frac{5}{6}, - \frac{1}{2}), a = \frac{13}{3}, b = \frac{13}{3}

Solution steps

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

Rewrite

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

in the form of the standard ellipse equation

\frac{( x - \frac{5}{6} ) ^{2}}{( \frac{13}{3} ) ^{2}} + \frac{( y - ( - \frac{1}{2} ) ) ^{2}}{( \frac{13}{3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{5}{6}, - \frac{1}{2}), a = \frac{13}{3}, b = \frac{13}{3}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{5}{6}, - \frac{1}{2}), b = \frac{13}{3}, a = \frac{13}{3}

\frac{( x - 3 ) ^{2}}{4} + \frac{( y - 3 ) ^{2}}{16} = 1

4 x^{2} + 2 y^{2} = 16

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

x^{2} + 4 y^{2} + 9 x + 9 y - 15 = 0

4 x^{2} + 16 y^{2} - 8 x + 96 y + 84 = 0

What is 3x^2+y^2-5x=2-y ?
The solution to 3x^2+y^2-5x=2-y is Ellipse with (h,k)=(5/6 ,-1/2),a=(sqrt(13))/3 ,b=sqrt(13/3)