What is 16x^2+16y^2=4y^2+20y+25 ?

The solution to 16x^2+16y^2=4y^2+20y+25 is Ellipse with (h,k)=(0, 5/6),a= 5/(2sqrt(3)),b= 5/3

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16x^2+16y^2=4y^2+20y+25

16 x^{2} + 16 y^{2} = 4 y^{2} + 20 y + 25

(h, k) = (0, \frac{5}{6}), a = \frac{5}{2 3}, b = \frac{5}{3}

Solution steps

16 x^{2} + 16 y^{2} = 4 y^{2} + 20 y + 25

Rewrite

16 x^{2} + 16 y^{2} - 4 y^{2} - 20 y - 25 = 0

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{( \frac{5}{2 3} ) ^{2}} + \frac{( y - \frac{5}{6} ) ^{2}}{( \frac{5}{3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, \frac{5}{6}), a = \frac{5}{2 3}, b = \frac{5}{3}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (0, \frac{5}{6}), b = \frac{5}{3}, a = \frac{5}{2 3}

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

4 x^{2} + 8 y^{2} - 8 x + 48 y + 44 = 0

\frac{( ( x - 5 ) ^{2} )}{9} + \frac{( ( y - 4 ) ^{2} )}{36} = 1

15 x^{2} + 10 y^{2} = 150

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

What is 16x^2+16y^2=4y^2+20y+25 ?
The solution to 16x^2+16y^2=4y^2+20y+25 is Ellipse with (h,k)=(0, 5/6),a= 5/(2sqrt(3)),b= 5/3