Let's solve the problem step-by-step. The given equation is:

$9x^2 + 16y^2 - 18x - 96y + 9 = 0.$

We will convert it into the standard form of an ellipse by completing the square for both $x$ and $y$.

Step 1: Group $x$-terms and $y$-terms

Reorganize the equation: $9(x^2 - 2x) + 16(y^2 - 6y) + 9 = 0.$

Step 2: Factor out coefficients of squared terms

Factor out 9 from the $x$-terms and 16 from the $y$-terms: $9\big(x^2 - 2x\big) + 16\big(y^2 - 6y\big) = -9.$

Step 3: Complete the square

For $x^2 - 2x$:

1. Take half of the coefficient of $x$, which is $-2$, and square it: $(\frac{-2}{2})^2 = 1$.
2. Add and subtract 1 inside the parentheses.

For $y^2 - 6y$:

1. Take half of the coefficient of $y$, which is $-6$, and square it: $(\frac{-6}{2})^2 = 9$.
2. Add and subtract 9 inside the parentheses.

The equation becomes: $9\big((x^2 - 2x + 1) - 1\big) + 16\big((y^2 - 6y + 9) - 9\big) = -9.$

Step 4: Simplify the squares

Simplify the completed squares: $9\big((x - 1)^2 - 1\big) + 16\big((y - 3)^2 - 9\big) = -9.$

Distribute the constants $9$ and $16$: $9(x - 1)^2 - 9 + 16(y - 3)^2 - 144 = -9.$

Simplify further: $9(x - 1)^2 + 16(y - 3)^2 - 153 = -9.$

Add 153 to both sides: $9(x - 1)^2 + 16(y - 3)^2 = 144.$

Step 5: Divide through by 144

To get the standard form, divide the entire equation by 144: $\frac{9(x - 1)^2}{144} + \frac{16(y - 3)^2}{144} = 1.$

Simplify the fractions: $\frac{(x - 1)^2}{16} + \frac{(y - 3)^2}{9} = 1.$

Final Answer:

The standard form of the ellipse is: $\boxed{\frac{(x - 1)^2}{16} + \frac{(y - 3)^2}{9} = 1.}$

This matches the first option in the given choices.

Do you want further explanation, or do you have additional questions?
Here are 5 related questions to explore:

1. How do you identify the center of an ellipse from its standard form?
2. How do the denominators in the standard form relate to the ellipse's axes?
3. What is the general process for completing the square?
4. How can you determine the orientation of the ellipse (horizontal or vertical)?
5. How do you find the foci of an ellipse given its equation?

Tip:

When completing the square, always remember to balance the equation by adding or subtracting the same values to both sides!