What is 3X^2+4Y^2+12X-8Y+9=0 ?

The solution to 3X^2+4Y^2+12X-8Y+9=0 is Ellipse with (h,k)=(-2,1),a=sqrt(7/3),b=(sqrt(7))/2

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3X^2+4Y^2+12X-8Y+9=0

3 X^{2} + 4 Y^{2} + 12 X - 8 Y + 9 = 0

(h, k) = (- 2, 1), a = \frac{7}{3}, b = \frac{7}{2}

Solution steps

3 X^{2} + 4 Y^{2} + 12 X - 8 Y + 9 = 0

Rewrite

3 X^{2} + 4 Y^{2} + 12 X - 8 Y + 9 = 0

in the form of the standard ellipse equation

\frac{( X - ( - 2 ) ) ^{2}}{( \frac{7}{3} ) ^{2}} + \frac{( Y - 1 ) ^{2}}{( \frac{7}{2} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (- 2, 1), a = \frac{7}{3}, b = \frac{7}{2}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (- 2, 1), a = \frac{7}{3}, b = \frac{7}{2}

\frac{( x + 2 ) ^{2}}{4} + \frac{( y - 3 ) ^{2}}{16} = 1

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

25 x^{2} + 4 y^{2} - 200 x + 24 y + 376 = 0

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

x^{2} + y^{2} - 2 y > 0

What is 3X^2+4Y^2+12X-8Y+9=0 ?
The solution to 3X^2+4Y^2+12X-8Y+9=0 is Ellipse with (h,k)=(-2,1),a=sqrt(7/3),b=(sqrt(7))/2