Question

How do you identity if the equation `25y^2+9x^2-50y-54x=119` is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Answer 1

The conic is an ellipse

Explanation:

We have:

`25y^2 + 9x^2 - 50y - 54x = 119`

First we can collect "like" terms:

`{9x^2 - 54x } + {25y^2 - 50y} = 119`

Next we complete the square independently on the `x` and `y` terms

`:. {9(x^2 - 54/9x) } + {25(y^2 - 50/25y)} = 119`

`:. 9{x^2 - 6x } + 25{y^2 - 2y} = 119`

`:. 9{(x-3)^2-3^2 } + 25{(y-1)^2-1^2} = 119`

`:. 9{(x-3)^2-9 } + 25{(y-1)^2-1} = 119`

We have now factorised the "quadratic" terms so we now collect constant terms:

`:. 9(x-3)^2 - 9*9 + 25(y-1)^2-25*1 = 119`

`:. 9(x-3)^2 + 25(y-1)^2 = 119 + 25 + 81`

`:. 9(x-3)^2 + 25(y-1)^2 = 225`

Finally, we put the equation into standard form:

`:. 9/225(x-3)^2 + 25/225(y-1)^2 = 1`

`:. 1/25(x-3)^2 + 1/9(y-1)^2 = 1`

`:. (x-3)^2/5^2 + (y-1)^2/3^2 = 1`

And as such, we can identify the conic as an ellipse

graph{25y^2 + 9x^2 - 50y - 54x = 119 [-10, 10, -5, 5]}