What is 3x^2+2y^2+6x-8y-25=0 ?

The solution to 3x^2+2y^2+6x-8y-25=0 is Ellipse with (h,k)=(-1,2),a=2sqrt(3),b=3sqrt(2)

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3x^2+2y^2+6x-8y-25=0

3 x^{2} + 2 y^{2} + 6 x - 8 y - 25 = 0

(h, k) = (- 1, 2), a = 23, b = 32

Solution steps

3 x^{2} + 2 y^{2} + 6 x - 8 y - 25 = 0

Rewrite

3 x^{2} + 2 y^{2} + 6 x - 8 y - 25 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (- 1, 2), a = 23, b = 32

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (- 1, 2), b = 32, a = 23

\frac{x ^{2}}{6 ^{2}} + \frac{y ^{2}}{3 ^{2}} = 1

\frac{( x + 5 ) ^{2}}{4} + \frac{( y - 1 ) ^{2}}{25} = 1

25 x^{2} + 36 y^{2} - 350 x + 350 = 0

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

15 x^{2} + 15 y^{2} + 16 y \geq 0

What is 3x^2+2y^2+6x-8y-25=0 ?
The solution to 3x^2+2y^2+6x-8y-25=0 is Ellipse with (h,k)=(-1,2),a=2sqrt(3),b=3sqrt(2)