### Instruction: Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty.

$x^2 + 2y^2 - 6x - 20y + 59 = 12$.### Response: To identify the type of conic section represented by the equation, I will try to rewrite it in standard form by completing the square for both x and y terms.
First, I will group the x terms and y terms together and move the constant term to the right side:

$(x^2 - 6x) + (2y^2 - 20y) = 12 - 59$
Then, I will factor out the coefficients of the squared terms:

$x^2 - 6x + (2)(y^2 - 10y) = -47$
Next, I will add the appropriate constants to both sides to complete the square for each variable:

$x^2 - 6x + 9 + (2)(y^2 - 10y + 25) = -47 + 9 + (2)(25)$
Simplifying the right side, I get:

$x^2 - 6x + 9 + (2)(y^2 - 10y + 25) = 12$
Then, I will factor the perfect square trinomials on the left side:

$(x - 3)^2 + 2(y - 5)^2 = 12$
Finally, I will divide both sides by 12 to get the equation in standard form:

$\frac{(x - 3)^2}{12} + \frac{(y - 5)^2}{6} = 1$
This is the standard form of an ellipse, where the center is at (3, 5), the major axis is horizontal, the semi-major axis is $\sqrt{12}$, and the semi-minor axis is $\sqrt{6}$.