What is 4x^2+3y^2+8x-24y+51=0 ?

The solution to 4x^2+3y^2+8x-24y+51=0 is Ellipse with (h,k)=(-1,4),a= 1/2 ,b=sqrt(1/3)

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4x^2+3y^2+8x-24y+51=0

4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0

(h, k) = (- 1, 4), a = \frac{1}{2}, b = \frac{1}{3}

Solution steps

4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0

Rewrite

4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (- 1, 4), a = \frac{1}{2}, b = \frac{1}{3}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (- 1, 4), b = \frac{1}{3}, a = \frac{1}{2}

9 y^{2} - 25 x^{2} = 225

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

\frac{x ^{2}}{81} + \frac{y ^{2}}{9} = 1

16 x^{2} + 16 y^{2} + 32 x + 8 y = 47

\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1

What is 4x^2+3y^2+8x-24y+51=0 ?
The solution to 4x^2+3y^2+8x-24y+51=0 is Ellipse with (h,k)=(-1,4),a= 1/2 ,b=sqrt(1/3)