What are the equations in standard form of the equations `9x^2+16y^2=144` and `25x^2+9y^2-18y-216=0` ?

Given: `9x^2 + 16y^2 = 144` and `25x^2 + 9y^2 -18y -216 = 0`

To point an ellipse or hyperbola in standard form you must complete the square if needed, then divide by the constant on the right side:

Equation 1: `" "(9x^2)/144 + (16y^2)/144 = 144/144`

`(9x^2)/144 + (16y^2)/144 = 1`

Reduce so there isn't a constant in the numerator:

`x^2/16 + y^2/9 = 1" "` horizontal ellipse centered at `(0, 0)`

It's horizontal because the largest denominator is on the `x`-term.

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Equation 2:
Requires completing of the square. First group `x` terms together, `y`-terms together and put the constant on the right:

`(25x^2) + (9y^2 -18y) = 216`

Factor: `25x^2 + 9(y^2 -2y) = 216`

Complete the square on `y` by halving the `-2y` constant `= -1` and adding the 9(-1)^2 on the right that was added to the left when the square was completed:

`25x^2 + 9(y-1)^2 = 216 + 9(-1)^2`

`25x^2 + 9(y-1)^2 = 225`

Divide by the constant on the right side:

`(25x^2)/225 + (9(y-1)^2)/225 = 225/225`

`(25x^2)/225 + (9(y-1)^2)/225 =1`

Reduce so there isn't a constant in the numerator:

`x^2/9 + (y-1)^2/25 =1" "` vertical ellipse centered at `(0, 1)`

It's vertical because the largest denominator is on the `y`-term.