ii. transformation of equations and identifying the properties based on equations.
fill in the table with the correct properties.

table:
column 1:

standard form |
general form |
orientation |
values of a, b, and c |
center | 
vertices |
covertices |
foci |
length of the major axis |
length of the minor axis |

column 2:
(x - 1)^2 + (y - 10)^2
-------------     -------------- = 1
   4                  3
^ answer to the standard form column.

column 3:

9x^2 + 16y^2 - 126x + 64y - 71 = 0 
^ answer to the general form.

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.

Explore the transformation of ellipse equations from standard to general form, and identify key properties such as orientation, vertices, covertices, and foci. Learn how to calculate values of a, b, and c, complete the square, and more. Suitable for Grades 10-12 students studying conic sections.

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