Let's work through the conversion of the given equation from general form to standard form step by step. The general form of the equation is:

$9x^2 + 4y^2 + 72x - 48y + 144 = 0$

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

We group the quadratic and linear terms involving $x$ and $y$:

$9(x^2 + 8x) + 4(y^2 - 12y) = -144$

Step 2: Complete the square.

Now, we complete the square for both the $x$-terms and the $y$-terms.

For the $x$-terms:

The expression is $x^2 + 8x$. To complete the square, take half of 8, which is 4, and then square it, giving 16. So:

$x^2 + 8x = (x + 4)^2 - 16$

For the $y$-terms:

The expression is $y^2 - 12y$. Take half of -12, which is -6, and square it to get 36. So:

$y^2 - 12y = (y - 6)^2 - 36$

Now substitute these into the equation:

$9((x + 4)^2 - 16) + 4((y - 6)^2 - 36) = -144$

Step 3: Simplify and move constants to the right-hand side.

Distribute the 9 and 4:

$9(x + 4)^2 - 144 + 4(y - 6)^2 - 144 = -144$

Combine like terms:

$9(x + 4)^2 + 4(y - 6)^2 - 288 = -144$

Add 288 to both sides:

$9(x + 4)^2 + 4(y - 6)^2 = 144$

Step 4: Divide by 144 to get the equation in standard form.

Divide everything by 144:

$\frac{9(x + 4)^2}{144} + \frac{4(y - 6)^2}{144} = 1$

Simplify the fractions:

$\frac{(x + 4)^2}{16} + \frac{(y - 6)^2}{36} = 1$

Step 5: Identify the values of $h$, $k$, $D$, and $E$.

From the standard form:

$\frac{(x - h)^2}{D} + \frac{(y - k)^2}{E} = 1$

we can compare with our result:

$\frac{(x + 4)^2}{16} + \frac{(y - 6)^2}{36} = 1$

Thus:

• $h = -4$
• $k = 6$
• $D = 16$
• $E = 36$

Would you like further details on any of the steps, or do you have any questions?

Related Questions:

1. How do you complete the square in quadratic expressions?
2. What is the process for converting a general conic equation to standard form?
3. What do $h$ and $k$ represent in the standard form of an ellipse?
4. How can you determine if a given conic equation represents a circle, ellipse, or hyperbola?
5. Why do we divide both sides of the equation by the constant term to standardize the form?

Tip:

Always double-check your completion of the square and constants when converting from general to standard form to avoid mistakes.