What is x^{(2)}+4y^{(2)}+7=12y-2x ?

The solution to x^{(2)}+4y^{(2)}+7=12y-2x is Ellipse with (h,k)=(-1, 3/2),a=sqrt(3),b=(sqrt(3))/2

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x^{(2)}+4y^{(2)}+7=12y-2x

x^{(2)} + 4 y^{(2)} + 7 = 12 y - 2 x

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

Solution steps

x^{2} + 4 y^{2} + 7 = 12 y - 2 x

Rewrite

x^{2} + 4 y^{2} + 7 = 12 y - 2 x

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

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

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

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

\frac{( x - 4 ) ^{2}}{4} + \frac{( y - 5 ) ^{2}}{36} = 1

\frac{( x - 1 ) ^{2}}{4 ^{2}} + \frac{( y - 5 ) ^{2}}{b ^{2}} = 1

x^{(2)} + 3 y^{(2)} - 6 x + 6 y = 0

5 x^{2} + 13 y^{2} - 65 = 0

y^{2} = 1 - 16 x^{2}

What is x^{(2)}+4y^{(2)}+7=12y-2x ?
The solution to x^{(2)}+4y^{(2)}+7=12y-2x is Ellipse with (h,k)=(-1, 3/2),a=sqrt(3),b=(sqrt(3))/2