What is y^2+4x^2+24x-16y=-84 ?

The solution to y^2+4x^2+24x-16y=-84 is Ellipse with (h,k)=(-3,8),a=2,b=4

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y^2+4x^2+24x-16y=-84

y^{2} + 4 x^{2} + 24 x - 16 y = - 84

(h, k) = (- 3, 8), a = 2, b = 4

Solution steps

y^{2} + 4 x^{2} + 24 x - 16 y = - 84

Rewrite

y^{2} + 4 x^{2} + 24 x - 16 y = - 84

in the form of the standard ellipse equation

\frac{( x - ( - 3 ) ) ^{2}}{2 ^{2}} + \frac{( y - 8 ) ^{2}}{4 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (- 3, 8), a = 2, b = 4

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (- 3, 8), b = 4, a = 2

9 x^{2} + 4 y^{2} + 54 x - 32 y + 109 = 0

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

\frac{x ^{2}}{8 ^{2}} + \frac{y ^{2}}{4 ^{2}} = 1

28 - 4 x^{2} - 7 y^{2} > 0

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

What is y^2+4x^2+24x-16y=-84 ?
The solution to y^2+4x^2+24x-16y=-84 is Ellipse with (h,k)=(-3,8),a=2,b=4