What is 6x^2+8y^2+12x-32y=178 ?

The solution to 6x^2+8y^2+12x-32y=178 is Ellipse with (h,k)=(-1,2),a=6,b=3sqrt(3)

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6x^2+8y^2+12x-32y=178

6 x^{2} + 8 y^{2} + 12 x - 32 y = 178

(h, k) = (- 1, 2), a = 6, b = 33

Solution steps

6 x^{2} + 8 y^{2} + 12 x - 32 y = 178

Rewrite

6 x^{2} + 8 y^{2} + 12 x - 32 y = 178

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (- 1, 2), a = 6, b = 33

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (- 1, 2), a = 6, b = 33

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

4 x^{2} + 9 y^{2} - 16 x + 18 y - 11 = 0

3 x^{2} + y^{2} - 24 x + 39 = 0

9 x^{2} + 16 y^{2} - 54 x + 64 y + 1 = 0

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

What is 6x^2+8y^2+12x-32y=178 ?
The solution to 6x^2+8y^2+12x-32y=178 is Ellipse with (h,k)=(-1,2),a=6,b=3sqrt(3)