What is x^2+(y-1)^2+(4+y)^2=64 ?

The solution to x^2+(y-1)^2+(4+y)^2=64 is Ellipse with (h,k)=(0,-3/2),a=sqrt(103/2),b=(sqrt(103))/2

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x^2+(y-1)^2+(4+y)^2=64

x^{2} + (y - 1)^{2} + (4 + y)^{2} = 64

(h, k) = (0, - \frac{3}{2}), a = \frac{103}{2}, b = \frac{103}{2}

Solution steps

x^{2} + (y - 1)^{2} + (4 + y)^{2} = 64

Rewrite

x^{2} + (y - 1)^{2} + (4 + y)^{2} = 64

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{( \frac{103}{2} ) ^{2}} + \frac{( y - ( - \frac{3}{2} ) ) ^{2}}{( \frac{103}{2} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, - \frac{3}{2}), a = \frac{103}{2}, b = \frac{103}{2}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (0, - \frac{3}{2}), a = \frac{103}{2}, b = \frac{103}{2}

\frac{( y + 1 ) ^{2}}{16} + \frac{x ^{2}}{4} - 1 = 0

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

\frac{x ^{2}}{8 ^{2}} + \frac{y ^{2}}{5 ^{2}} = 1

x^{2} - 4 x + 4 y^{2} + 4 y + 1 = 0

25 x^{2} + 49 y^{2} = 49

What is x^2+(y-1)^2+(4+y)^2=64 ?
The solution to x^2+(y-1)^2+(4+y)^2=64 is Ellipse with (h,k)=(0,-3/2),a=sqrt(103/2),b=(sqrt(103))/2