Find the Center 81x^2+36y^2+972x-648y+2916=0

Solution:

Step:1

Convert the given equation into the standard form of an ellipse.

Step:1.1

Subtract $2916$ on both sides to isolate the variable terms.

$$81x^2 + 36y^2 + 972x - 648y = -2916$$

Step:1.2

Complete the square for the $x$-terms, $81x^2 + 972x$.

Step:1.2.1

Identify coefficients for $ax^2 + bx + c$ where $a=81$, $b=972$, and $c=0$.

Step:1.2.2

Use the vertex form $a(x + d)^2 + e$ for a parabola.

Step:1.2.3

Calculate $d$ with $d = \frac{b}{2a}$.

Step:1.2.3.1

Insert $a$ and $b$ into the formula.

$$d = \frac{972}{2 \cdot 81}$$

Step:1.2.3.2

Simplify the fraction by canceling common factors.

Step:1.2.3.2.1

Factor out a $2$ from $972$.

$$d = \frac{2 \cdot 486}{2 \cdot 81}$$

Step:1.2.3.2.2

Remove the common factor of $2$.

$$d = \frac{\cancel{2} \cdot 486}{\cancel{2} \cdot 81}$$

Step:1.2.3.2.3

Simplify the fraction.

$$d = \frac{486}{81}$$

Step:1.2.3.2.4

Reduce the fraction by canceling the common factor of $486$ and $81$.

$$d = \frac{6}{1}$$ $$d = 6$$

Step:1.2.4

Determine $e$ with $e = c - \frac{b^2}{4a}$.

Step:1.2.4.1

Plug in $c$, $b$, and $a$.

$$e = 0 - \frac{972^2}{4 \cdot 81}$$

Step:1.2.4.2

Simplify the expression.

Step:1.2.4.2.1

Perform the arithmetic operations.

$$e = 0 - \frac{944784}{324}$$ $$e = -2916$$

Step:1.2.5

Insert $a$, $d$, and $e$ into the vertex form.

$$81(x + 6)^2 - 2916$$

Step:1.3

Replace $81x^2 + 972x$ with $81(x + 6)^2 - 2916$ in the original equation.

$$81(x + 6)^2 + 36y^2 - 648y = -2916$$

Step:1.4

Cancel out $-2916$ on both sides.

$$81(x + 6)^2 + 36y^2 - 648y = 0$$

Step:1.5

Complete the square for the $y$-terms, $36y^2 - 648y$.

Step:1.5.1

Identify coefficients for $ay^2 + by + c$ where $a=36$, $b=-648$, and $c=0$.

Step:1.5.2

Use the vertex form $a(y + d)^2 + e$ for a parabola.

Step:1.5.3

Calculate $d$ with $d = \frac{b}{2a}$.

Step:1.5.3.1

Insert $a$ and $b$ into the formula.

$$d = \frac{-648}{2 \cdot 36}$$

Step:1.5.3.2

Simplify the fraction by canceling common factors.

Step:1.5.3.2.1

Factor out a $2$ from $-648$.

$$d = \frac{2 \cdot -324}{2 \cdot 36}$$

Step:1.5.3.2.2

Remove the common factor of $2$.

$$d = \frac{\cancel{2} \cdot -324}{\cancel{2} \cdot 36}$$

Step:1.5.3.2.3

Simplify the fraction.

$$d = \frac{-324}{36}$$

Step:1.5.3.2.4

Reduce the fraction by canceling the common factor of $-324$ and $36$.

$$d = \frac{-9}{1}$$ $$d = -9$$

Step:1.5.4

Determine $e$ with $e = c - \frac{b^2}{4a}$.

Step:1.5.4.1

Plug in $c$, $b$, and $a$.

$$e = 0 - \frac{(-648)^2}{4 \cdot 36}$$

Step:1.5.4.2

Simplify the expression.

$$e = -2916$$

Step:1.5.5

Insert $a$, $d$, and $e$ into the vertex form.

$$36(y - 9)^2 - 2916$$

Step:1.6

Replace $36y^2 - 648y$ with $36(y - 9)^2 - 2916$ in the original equation.

$$81(x + 6)^2 + 36(y - 9)^2 = 0$$

Step:1.7

Eliminate $-2916$ by adding $2916$ to both sides.

$$81(x + 6)^2 + 36(y - 9)^2 = 2916$$

Step:1.8

Simplify the equation to get the standard form.

$$\frac{(x + 6)^2}{36} + \frac{(y - 9)^2}{81} = 1$$

Step:2

Recognize that the equation is now in the standard form of an ellipse.

Step:3

Identify the center and axes lengths from the standard form.

$$a = 9, b = 6, h = -6, k = 9$$

Step:4

The center of the ellipse is given by the coordinates $(h, k)$.

$$\text{Center} = (-6, 9)$$

Knowledge Notes:

To solve for the center of an ellipse given in the general quadratic form, one must first rewrite the equation in the standard form of an ellipse. The standard form for an ellipse is:

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$ where $(h, k)$ is the center of the ellipse, $a$ is the semi-major axis, and $b$ is the semi-minor axis.

The process involves:

1. Rearranging the equation to group the $x$ and $y$ terms.

2. Completing the square for both $x$ and $y$ terms to isolate them on one side.

3. Dividing through by the constant term to normalize the equation to equal $1$.

4. Comparing the resulting equation to the standard form to identify the center and axes lengths.

The vertex form of a parabola is used to complete the square, which is $a(x + d)^2 + e$ for $x$-terms and $a(y + d)^2 + e$ for $y$-terms. The value of $d$ is found using the formula $d = -\frac{b}{2a}$, and $e$ is found using $e = c - \frac{b^2}{4a}$, where $a$, $b$, and $c$ are coefficients from the general quadratic form $ax^2 + bx + c$.