What is 2x^2+y^2-2x+16y+1=0 ?

The solution to 2x^2+y^2-2x+16y+1=0 is Ellipse with (h,k)=(1/2 ,-8),a=(sqrt(127))/2 ,b=sqrt(127/2)

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2x^2+y^2-2x+16y+1=0

2 x^{2} + y^{2} - 2 x + 16 y + 1 = 0

(h, k) = (\frac{1}{2}, - 8), a = \frac{127}{2}, b = \frac{127}{2}

Solution steps

2 x^{2} + y^{2} - 2 x + 16 y + 1 = 0

Rewrite

2 x^{2} + y^{2} - 2 x + 16 y + 1 = 0

in the form of the standard ellipse equation

\frac{( x - \frac{1}{2} ) ^{2}}{( \frac{127}{2} ) ^{2}} + \frac{( y - ( - 8 ) ) ^{2}}{( \frac{127}{2} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{2}, - 8), a = \frac{127}{2}, b = \frac{127}{2}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{2}, - 8), b = \frac{127}{2}, a = \frac{127}{2}

x^{2} + y^{2} + 2 y \leq 0

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

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

\frac{( x + 2 ) ^{2}}{16} + \frac{( y - 2 ) ^{2}}{12} = 1

2 x^{2} + y^{2} + 10 x + 12 y = - 12

What is 2x^2+y^2-2x+16y+1=0 ?
The solution to 2x^2+y^2-2x+16y+1=0 is Ellipse with (h,k)=(1/2 ,-8),a=(sqrt(127))/2 ,b=sqrt(127/2)