What is 4x^2+9y^2-48x+72y=-144 ?

The solution to 4x^2+9y^2-48x+72y=-144 is Ellipse with (h,k)=(6,-4),a=6,b=4

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4x^2+9y^2-48x+72y=-144

4 x^{2} + 9 y^{2} - 48 x + 72 y = - 144

(h, k) = (6, - 4), a = 6, b = 4

Solution steps

4 x^{2} + 9 y^{2} - 48 x + 72 y = - 144

Rewrite

4 x^{2} + 9 y^{2} - 48 x + 72 y = - 144

in the form of the standard ellipse equation

\frac{( x - 6 ) ^{2}}{6 ^{2}} + \frac{( y - ( - 4 ) ) ^{2}}{4 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (6, - 4), a = 6, b = 4

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (6, - 4), a = 6, b = 4

4 - x^{2} - 4 y^{2} \geq 0

\frac{( x + 5 ) ^{2}}{9} + \frac{( y - 4 ) ^{2}}{36} = 1

\frac{x ^{2}}{9} + \frac{y ^{2}}{45} = 1

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

x^{2} + 4 y^{2} - 8 y = 0

What is 4x^2+9y^2-48x+72y=-144 ?
The solution to 4x^2+9y^2-48x+72y=-144 is Ellipse with (h,k)=(6,-4),a=6,b=4