What is x^2+10x+3y^2+10y=14 ?

The solution to x^2+10x+3y^2+10y=14 is Ellipse with (h,k)=(-5,-5/3),a=sqrt(142/3),b=(sqrt(142))/3

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x^2+10x+3y^2+10y=14

x^{2} + 10 x + 3 y^{2} + 10 y = 14

(h, k) = (- 5, - \frac{5}{3}), a = \frac{142}{3}, b = \frac{142}{3}

Solution steps

x^{2} + 10 x + 3 y^{2} + 10 y = 14

Rewrite

x^{2} + 10 x + 3 y^{2} + 10 y = 14

in the form of the standard ellipse equation

\frac{( x - ( - 5 ) ) ^{2}}{( \frac{142}{3} ) ^{2}} + \frac{( y - ( - \frac{5}{3} ) ) ^{2}}{( \frac{142}{3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (- 5, - \frac{5}{3}), a = \frac{142}{3}, b = \frac{142}{3}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (- 5, - \frac{5}{3}), a = \frac{142}{3}, b = \frac{142}{3}

\frac{x ^{2}}{12} + \frac{y ^{2}}{12 - 16} = 1

22 x^{2} + 4 y^{2} = 1

x^{2} + 4 y^{2} + 10 x + 9 y - 10 = 0

16 x^{2} + 81 y^{2} = 81

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

What is x^2+10x+3y^2+10y=14 ?
The solution to x^2+10x+3y^2+10y=14 is Ellipse with (h,k)=(-5,-5/3),a=sqrt(142/3),b=(sqrt(142))/3