Let's fill in the table with the correct properties of the given equations. The first equation represents the standard form of an ellipse, while the second is the general form.

Given Equations:

1. Standard Form:
   $\frac{(x - 1)^2}{4} + \frac{(y - 10)^2}{3} = 1$

2. General Form:
   $9x^2 + 16y^2 - 126x + 64y - 71 = 0$

Table with Properties:

Property	Equation 1 (Standard Form)	Equation 2 (General Form)
Standard Form	$\frac{(x - 1)^2}{4} + \frac{(y - 10)^2}{3} = 1$	-
General Form	-	$9x^2 + 16y^2 - 126x + 64y - 71 = 0$
Orientation	Horizontal (because $a^2 > b^2$)	Horizontal
Values of a, b, and c	$a = 2$, $b = \sqrt{3} \approx 1.732$, $c = \sqrt{a^2 - b^2} = 1$	Found after completing the square
Center	$(1, 10)$	$(7, -2)$ (after completing the square)
Vertices	$(1 \pm 2, 10) = (-1, 10)$ and $(3, 10)$	Found after converting to standard form
Covertices	$(1, 10 \pm \sqrt{3}) = (1, 10 \pm 1.732)$	Found after converting to standard form
Foci	$(1 \pm 1, 10) = (0, 10)$ and $(2, 10)$	Found after converting to standard form
Length of the Major Axis	$2a = 4$	Found after converting to standard form
Length of the Minor Axis	$2b = 2\sqrt{3} \approx 3.464$	Found after converting to standard form

Key Steps for Equation 2:

For the general form $9x^2 + 16y^2 - 126x + 64y - 71 = 0$, we would need to complete the square and rearrange it into the standard form of an ellipse. This will give us the center, vertices, covertices, and foci.

Would you like to see the detailed steps of converting the general form into standard form?

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Follow-up Questions:

1. What is the process of completing the square for the general form of a conic?
2. How do you calculate the values of $a$, $b$, and $c$ for ellipses?
3. Why does the major axis length depend on $a$?
4. How do we determine if an ellipse is horizontally or vertically oriented?
5. What is the geometric interpretation of the foci of an ellipse?

Tip: In ellipses, the larger denominator under the squared terms indicates the direction of the major axis.