What is x^2-xy+1/2 (x+y)^2=4 ?

The solution to x^2-xy+1/2 (x+y)^2=4 is Ellipse with (h,k)=(0,0),a=(2sqrt(2))/(sqrt(3)),b=2sqrt(2)

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x^2-xy+1/2 (x+y)^2=4

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

(h, k) = (0, 0), a = \frac{2 2}{3}, b = 22

Solution steps

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

Rewrite

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

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (0, 0), a = \frac{2 2}{3}, b = 22

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (0, 0), b = 22, a = \frac{2 2}{3}

(4 x^{2} - 8 x) + (9 y^{2} - 36 y) = - 4

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

\frac{x ^{2}}{1 ^{2}} + \frac{y ^{2}}{0.4375 ^{2}} = 1

22 x^{2} + 66 y^{2} + 60 x + 4 y - 9 = 0

x^{2} + 2 y^{2} - 5 = 1

What is x^2-xy+1/2 (x+y)^2=4 ?
The solution to x^2-xy+1/2 (x+y)^2=4 is Ellipse with (h,k)=(0,0),a=(2sqrt(2))/(sqrt(3)),b=2sqrt(2)