What is 4x^2+6y^2+4x-24y=24 ?

The solution to 4x^2+6y^2+4x-24y=24 is Ellipse with (h,k)=(-1/2 ,2),a= 7/2 ,b= 7/(sqrt(6))

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4x^2+6y^2+4x-24y=24

4 x^{2} + 6 y^{2} + 4 x - 24 y = 24

(h, k) = (- \frac{1}{2}, 2), a = \frac{7}{2}, b = \frac{7}{6}

Solution steps

4 x^{2} + 6 y^{2} + 4 x - 24 y = 24

Rewrite

4 x^{2} + 6 y^{2} + 4 x - 24 y = 24

in the form of the standard ellipse equation

\frac{( x - ( - \frac{1}{2} ) ) ^{2}}{( \frac{7}{2} ) ^{2}} + \frac{( y - 2 ) ^{2}}{( \frac{7}{6} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (- \frac{1}{2}, 2), a = \frac{7}{2}, b = \frac{7}{6}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (- \frac{1}{2}, 2), a = \frac{7}{2}, b = \frac{7}{6}

x^{2} + 2 y^{2} = 5

2 x^{2} + 4 y^{2} = 5

9 x^{2} + 4 y^{2} \leq 36

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

\frac{x ^{2}}{36} + \frac{y ^{2}}{20} = 1

What is 4x^2+6y^2+4x-24y=24 ?
The solution to 4x^2+6y^2+4x-24y=24 is Ellipse with (h,k)=(-1/2 ,2),a= 7/2 ,b= 7/(sqrt(6))