To write the equation of the ellipse in standard form and identify its center and vertices, we need to complete the square for both the $x$ and $y$ terms in the given equation:

$9x^2 + 18x + 4y^2 - 8y - 23 = 0$

First, group the $x$ and $y$ terms:

$9(x^2 + 2x) + 4(y^2 - 2y) - 23 = 0$

Next, complete the square for the $x$ terms:

1. Factor out the coefficient of $x^2$:

$9(x^2 + 2x) = 9\left(x^2 + 2x + 1 - 1\right) = 9\left((x + 1)^2 - 1\right) = 9(x + 1)^2 - 9$

2. Similarly, complete the square for the $y$ terms:

$4(y^2 - 2y) = 4\left(y^2 - 2y + 1 - 1\right) = 4\left((y - 1)^2 - 1\right) = 4(y - 1)^2 - 4$

Now substitute these completed squares back into the original equation:

$9(x + 1)^2 - 9 + 4(y - 1)^2 - 4 - 23 = 0$

Combine like terms:

$9(x + 1)^2 + 4(y - 1)^2 - 36 = 0$

Move the constant term to the other side of the equation:

$9(x + 1)^2 + 4(y - 1)^2 = 36$

Divide by 36 to normalize the equation:

$\frac{9(x + 1)^2}{36} + \frac{4(y - 1)^2}{36} = 1$

Simplify the fractions:

$\frac{(x + 1)^2}{4} + \frac{(y - 1)^2}{9} = 1$

This is the standard form of the ellipse equation.

Center and Vertices:

• The center $(h, k)$ is $(-1, 1)$.

To find the vertices, identify the lengths of the semi-major axis $a$ and the semi-minor axis $b$:

• $a^2 = 9$ so $a = 3$ (since 9 is under the $y$ term, the major axis is vertical).
• $b^2 = 4$ so $b = 2$.

Vertices:

Since the major axis is vertical, the vertices are 3 units above and below the center:

• Vertices: $(-1, 1 \pm 3)$ which are $(-1, 4)$ and $(-1, -2)$.

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the equation of an ellipse from the general conic equation?
2. What are the steps to complete the square for quadratic equations?
3. How can you identify the major and minor axes of an ellipse?
4. What are the foci of the ellipse, and how do you find their coordinates?
5. How do you graph an ellipse given its standard form equation?
6. What is the significance of the semi-major and semi-minor axes in an ellipse?
7. How can you verify if a given equation represents an ellipse?
8. What are the properties of ellipses that differentiate them from other conic sections?

Tip: To quickly determine if a conic section equation represents an ellipse, check if the coefficients of $x^2$ and $y^2$ are both positive and unequal.