What is 6x^2-30x+3y^2+30y+111=0 ?

The solution to 6x^2-30x+3y^2+30y+111=0 is Ellipse with (h,k)=(5/2 ,-5),a= 1/2 ,b=sqrt(1/2)

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6x^2-30x+3y^2+30y+111=0

6 x^{2} - 30 x + 3 y^{2} + 30 y + 111 = 0

(h, k) = (\frac{5}{2}, - 5), a = \frac{1}{2}, b = \frac{1}{2}

Solution steps

6 x^{2} - 30 x + 3 y^{2} + 30 y + 111 = 0

Rewrite

6 x^{2} - 30 x + 3 y^{2} + 30 y + 111 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (\frac{5}{2}, - 5), a = \frac{1}{2}, b = \frac{1}{2}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{5}{2}, - 5), b = \frac{1}{2}, a = \frac{1}{2}

13 x^{2} + 23 y^{2} - 51 x - 19 y - 4 = 0

250 = 2 x^{2} + \frac{y ^{2}}{2}

6 x^{2} + 18 y^{2} + 24 x = 0

x^{2} + 2 y^{2} - 6 = 0

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

What is 6x^2-30x+3y^2+30y+111=0 ?
The solution to 6x^2-30x+3y^2+30y+111=0 is Ellipse with (h,k)=(5/2 ,-5),a= 1/2 ,b=sqrt(1/2)