Question

How do you know if the conic section `3x^2 +6x + 5y^2 -20y- 13= 0` is a parabola, an ellipse, a hyperbola, or a circle?

Answer 1

Ellipse

Explanation:

We have:

`3x^2 +6x + 5y^2 -20y- 13 = 0`

To classify the conic, we collect terms in `x` and `y` and complete the square on those terms:

`3{x^2 +2x} + 5{y^2 -4y} - 13 = 0`

`:. 3{(x+1)^2-1^2 } + 5{(y-2)^2-(-2)^2} - 13 = 0`

`:. 3(x+1)^2-3 + 5(y-2)^2-20 - 13 = 0`

`:. 3(x+1)^2 + 5(y-2)^2 = 36`

`:. (3(x+1)^2)/36 + (5(y-2)^2)/36 = 1`

`:. (x+1)^2/12 + (y-2)^2/(36/5) = 1`

Which is the equation of a ellipse in standard form.

And, we can verify this graphically:
graph{3x^2 +6x + 5y^2 -20y- 13= 0 [-10, 10, -5, 5]}