What is 12x^2+96x+5y^2+50y+257=0 ?

The solution to 12x^2+96x+5y^2+50y+257=0 is Ellipse with (h,k)=(-4,-5),a=sqrt(5),b=2sqrt(3)

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12x^2+96x+5y^2+50y+257=0

12 x^{2} + 96 x + 5 y^{2} + 50 y + 257 = 0

(h, k) = (- 4, - 5), a = 5, b = 23

Solution steps

12 x^{2} + 96 x + 5 y^{2} + 50 y + 257 = 0

Rewrite

12 x^{2} + 96 x + 5 y^{2} + 50 y + 257 = 0

in the form of the standard ellipse equation

\frac{( x - ( - 4 ) ) ^{2}}{( 5 ) ^{2}} + \frac{( y - ( - 5 ) ) ^{2}}{( 2 3 ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (- 4, - 5), a = 5, b = 23

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (- 4, - 5), b = 23, a = 5

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

\frac{( x - 2 ) ^{2}}{81} + \frac{( y + 5 ) ^{2}}{64} = 1

\frac{( x + 2 ) ^{2}}{9} + \frac{( y - 3 ) ^{2}}{25} = 1

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

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

What is 12x^2+96x+5y^2+50y+257=0 ?
The solution to 12x^2+96x+5y^2+50y+257=0 is Ellipse with (h,k)=(-4,-5),a=sqrt(5),b=2sqrt(3)