Let's solve these questions step by step.

1. Matching the standard form of the conic section

To determine the type of conic section, we analyze each equation:

1. Equation 1: $4x^2 + 9y^2 + 40x - 36y - 8 = 0$

   • The coefficients of $x^2$ and $y^2$ are positive but different ($4$ and $9$), which indicates this is an ellipse.
   • Match this to B. Ellipse.

2. Equation 2: $4x - 7y + 23 = 10$

   • Simplify: $4x - 7y + 13 = 0$.
   • There are no squared terms ($x^2$ or $y^2$). This is a linear equation and therefore not a conic.
   • Match this to C. Not a Conic.

3. Equation 3: $x^2 + y^2 - 4x - 18y + 49 = 0$

   • Rearrange terms to complete the square:
     $x^2 - 4x + y^2 - 18y + 49 = 0$.
     Completing the square for $x$ and $y$:
     • $x^2 - 4x = (x-2)^2 - 4$,
     • $y^2 - 18y = (y-9)^2 - 81$.
       Substituting back:
       $(x-2)^2 - 4 + (y-9)^2 - 81 + 49 = 0 \implies (x-2)^2 + (y-9)^2 = 36.$
       This is a circle with center $(2, 9)$ and radius $6$.
     • Match this to A. Circle.

Final Matching:

1. B. Ellipse
2. C. Not a Conic
3. A. Circle

2. Writing the equation of the ellipse

Vertices: $(-1, 4)$ and $(-1, -6)$
Foci: $(-1, 3)$ and $(-1, -5)$

Step 1: Center of the ellipse
The center lies midway between the vertices.
$\text{Center} = \left( -1, \frac{4 + (-6)}{2} \right) = (-1, -1).$

Step 2: Distance between vertices (major axis length)
The distance between the vertices is:
$|4 - (-6)| = 10.$
Thus, the semi-major axis length is:
$a = \frac{10}{2} = 5.$

Step 3: Distance between foci
The distance between the foci is:
$|3 - (-5)| = 8.$
Thus, the distance from the center to a focus (c) is:
$c = \frac{8}{2} = 4.$

Step 4: Semi-minor axis length ($b$)
Using the ellipse relation $a^2 = b^2 + c^2$:
$5^2 = b^2 + 4^2 \implies 25 = b^2 + 16 \implies b^2 = 9 \implies b = 3.$

Step 5: Equation of the ellipse
The major axis is vertical (aligned along $y$) because the vertices have the same $x$-coordinate. Therefore, the equation is:
$\frac{(x+1)^2}{b^2} + \frac{(y+1)^2}{a^2} = 1.$
Substitute $a^2 = 25$ and $b^2 = 9$:
$\frac{(x+1)^2}{9} + \frac{(y+1)^2}{25} = 1.$

Correct Answer:
$\frac{(x+1)^2}{9} + \frac{(y+1)^2}{25} = 1.$

Final Results:

1. Matching the Conic Sections:

   • 1: B. Ellipse
   • 2: C. Not a Conic
   • 3: A. Circle

2. Equation of the Ellipse:
   $\frac{(x+1)^2}{9} + \frac{(y+1)^2}{25} = 1.$

Follow-Up Questions:

1. How do you complete the square for conic equations?
2. What is the general form of an ellipse centered at $(h, k)$?
3. How do you determine the major and minor axes of an ellipse?
4. What distinguishes a circle from an ellipse mathematically?
5. How do you identify whether an equation represents a conic section?

Tip:

To identify conic sections, focus on the coefficients of $x^2$ and $y^2$. If they are equal and positive, it's a circle; if they differ but have the same sign, it's an ellipse.