What is 4x^2+3y^2+4x=1 ?

The solution to 4x^2+3y^2+4x=1 is Ellipse with (h,k)=(-1/2 ,0),a=sqrt(1/2),b=sqrt(2/3)

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4x^2+3y^2+4x=1

4 x^{2} + 3 y^{2} + 4 x = 1

(h, k) = (- \frac{1}{2}, 0), a = \frac{1}{2}, b = \frac{2}{3}

Solution steps

4 x^{2} + 3 y^{2} + 4 x = 1

Rewrite

4 x^{2} + 3 y^{2} + 4 x = 1

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (- \frac{1}{2}, 0), a = \frac{1}{2}, b = \frac{2}{3}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (- \frac{1}{2}, 0), b = \frac{2}{3}, a = \frac{1}{2}

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

100 x^{2} + 9 y^{2} = 900

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

\frac{( x - 2 ) ^{2}}{49} + \frac{( y + 4 ) ^{2}}{16} = 1

25 x^{2} + 4 y^{2} - 225 = 0

What is 4x^2+3y^2+4x=1 ?
The solution to 4x^2+3y^2+4x=1 is Ellipse with (h,k)=(-1/2 ,0),a=sqrt(1/2),b=sqrt(2/3)