What is 4x^2-40x+9y^2+54y+145=0 ?

The solution to 4x^2-40x+9y^2+54y+145=0 is Ellipse with (h,k)=(5,-3),a=3,b=2

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4x^2-40x+9y^2+54y+145=0

4 x^{2} - 40 x + 9 y^{2} + 54 y + 145 = 0

(h, k) = (5, - 3), a = 3, b = 2

Solution steps

4 x^{2} - 40 x + 9 y^{2} + 54 y + 145 = 0

Rewrite

4 x^{2} - 40 x + 9 y^{2} + 54 y + 145 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (5, - 3), a = 3, b = 2

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (5, - 3), a = 3, b = 2

9 X^{2} + Y^{2} = 81

\frac{( x ^{2} - 2 x + 1 )}{34} + \frac{( y ^{2} - 8 y + 16 )}{25} = 1

y^{2} \leq - 4 x^{2} + 4

\frac{( x - 4 ) ^{2}}{35} + \frac{( y - 2 ) ^{2}}{39} = 1

16 - 4 x^{2} - y^{2} > 0

What is 4x^2-40x+9y^2+54y+145=0 ?
The solution to 4x^2-40x+9y^2+54y+145=0 is Ellipse with (h,k)=(5,-3),a=3,b=2