What is 3(3x^{(2)}+9y^{(2)}-2x-36)=-82 ?

The solution to 3(3x^{(2)}+9y^{(2)}-2x-36)=-82 is Ellipse with (h,k)=(1/3 ,0),a=sqrt(3),b=1

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3(3x^{(2)}+9y^{(2)}-2x-36)=-82

3 (3 x^{(2)} + 9 y^{(2)} - 2 x - 36) = - 82

(h, k) = (\frac{1}{3}, 0), a = 3, b = 1

Solution steps

3 (3 x^{2} + 9 y^{2} - 2 x - 36) = - 82

Rewrite

3 (3 x^{2} + 9 y^{2} - 2 x - 36) = - 82

in the form of the standard ellipse equation

\frac{( x - \frac{1}{3} ) ^{2}}{( 3 ) ^{2}} + \frac{( y - 0 ) ^{2}}{1 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{3}, 0), a = 3, b = 1

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{3}, 0), a = 3, b = 1

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

25 (x^{2} + 4 x + 4) + 36 (y^{2} + 2 y + 1) = 770

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

9 x^{2} - 18 x + 4 y^{2} + 24 y = - 9

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

What is 3(3x^{(2)}+9y^{(2)}-2x-36)=-82 ?
The solution to 3(3x^{(2)}+9y^{(2)}-2x-36)=-82 is Ellipse with (h,k)=(1/3 ,0),a=sqrt(3),b=1