What is x^2-x+4y^2+24y+30=4 ?

The solution to x^2-x+4y^2+24y+30=4 is Ellipse with (h,k)=(1/2 ,-3),a=(sqrt(41))/2 ,b=(sqrt(41))/4

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x^2-x+4y^2+24y+30=4

x^{2} - x + 4 y^{2} + 24 y + 30 = 4

(h, k) = (\frac{1}{2}, - 3), a = \frac{41}{2}, b = \frac{41}{4}

Solution steps

x^{2} - x + 4 y^{2} + 24 y + 30 = 4

Rewrite

x^{2} - x + 4 y^{2} + 24 y + 30 = 4

in the form of the standard ellipse equation

\frac{( x - \frac{1}{2} ) ^{2}}{( \frac{41}{2} ) ^{2}} + \frac{( y - ( - 3 ) ) ^{2}}{( \frac{41}{4} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{2}, - 3), a = \frac{41}{2}, b = \frac{41}{4}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{2}, - 3), a = \frac{41}{2}, b = \frac{41}{4}

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

5 x^{2} + 9 y^{2} - 8 x - 4 = 0

\frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1

10 x^{2} + 6 y^{2} - 12 x + 8 y - 15 = 0

What is x^2-x+4y^2+24y+30=4 ?
The solution to x^2-x+4y^2+24y+30=4 is Ellipse with (h,k)=(1/2 ,-3),a=(sqrt(41))/2 ,b=(sqrt(41))/4