# 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?

Apr 28, 2018

Ellipse

#### Explanation:

We have:

$3 {x}^{2} + 6 x + 5 {y}^{2} - 20 y - 13 = 0$

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

$3 \left\{{x}^{2} + 2 x\right\} + 5 \left\{{y}^{2} - 4 y\right\} - 13 = 0$

$\therefore 3 \left\{{\left(x + 1\right)}^{2} - {1}^{2}\right\} + 5 \left\{{\left(y - 2\right)}^{2} - {\left(- 2\right)}^{2}\right\} - 13 = 0$

$\therefore 3 {\left(x + 1\right)}^{2} - 3 + 5 {\left(y - 2\right)}^{2} - 20 - 13 = 0$

$\therefore 3 {\left(x + 1\right)}^{2} + 5 {\left(y - 2\right)}^{2} = 36$

$\therefore \frac{3 {\left(x + 1\right)}^{2}}{36} + \frac{5 {\left(y - 2\right)}^{2}}{36} = 1$

$\therefore {\left(x + 1\right)}^{2} / 12 + {\left(y - 2\right)}^{2} / \left(\frac{36}{5}\right) = 1$

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]}