Find the Center x^2+6x+9x^2-36y+9=0

Solution:

Step 1: Convert the equation to vertex form

Step 1.1: Isolate the variable $y$

Step 1.1.1: Combine like terms $x^2$ and $9x^2$ to get $10x^2 + 6x - 36y + 9 = 0$

Step 1.1.2: Rearrange the equation to move non-$y$ terms to the other side

Step 1.1.2.1: Subtract $10x^2$ to obtain $6x - 36y + 9 = -10x^2$

Step 1.1.2.2: Subtract $6x$ to get $-36y + 9 = -10x^2 - 6x$

Step 1.1.2.3: Subtract $9$ to have $-36y = -10x^2 - 6x - 9$

Step 1.1.3: Divide each term by $-36$ to isolate $y$

Step 1.1.3.1: Perform the division to get $y = \frac{10x^2}{36} + \frac{6x}{36} + \frac{9}{36}$

Step 1.1.3.2: Simplify the equation

Step 1.1.3.2.1: Reduce fractions to simplest form

Step 1.1.3.2.2: The simplified form is $y = \frac{5x^2}{18} + \frac{x}{6} + \frac{1}{4}$

Step 1.2: Complete the square for the $x$-terms

Step 1.2.1: Identify coefficients $a = \frac{5}{18}$, $b = \frac{1}{6}$, and $c = \frac{1}{4}$

Step 1.2.2: Use the vertex form $a(x + d)^2 + e$

Step 1.2.3: Calculate $d = \frac{b}{2a}$

Step 1.2.3.1: Insert values into the formula to get $d = \frac{1/6}{2(5/18)}$

Step 1.2.3.2: Simplify to find $d = \frac{3}{10}$

Step 1.2.4: Calculate $e = c - \frac{b^2}{4a}$

Step 1.2.4.1: Insert values into the formula to get $e = \frac{1}{4} - \frac{(1/6)^2}{4(5/18)}$

Step 1.2.4.2: Simplify to find $e = \frac{9}{40}$

Step 1.2.5: Substitute $a$, $d$, and $e$ into the vertex form to get $y = \frac{5}{18}(x + \frac{3}{10})^2 + \frac{9}{40}$

Step 1.3: Set $y$ to the new expression to finalize the vertex form

Step 2: Identify the coefficients in the vertex form

$a = \frac{5}{18}$, $h = -\frac{3}{10}$, $k = \frac{9}{40}$

Step 3: Determine the direction of the parabola

Since $a$ is positive, the parabola opens upwards

Step 4: Find the vertex $(h, k)$

Vertex is $(-\frac{3}{10}, \frac{9}{40})$

Step 5: Calculate the distance $p$ from the vertex to the focus

Step 5.1: Use the formula $p = \frac{1}{4a}$

Step 5.2: Insert $a$ into the formula to get $p = \frac{1}{4(\frac{5}{18})}$

Step 5.3: Simplify to find $p = \frac{9}{10}$

Step 6: Locate the focus of the parabola

Step 6.1: Add $p$ to the $k$ to find the focus if the parabola opens up

Step 6.2: Focus is $(-\frac{3}{10}, \frac{9}{40} + \frac{9}{10})$

Step 7: Determine the axis of symmetry

Axis of symmetry is $x = -\frac{3}{10}$

Step 8: Find the directrix of the parabola

Step 8.1: Subtract $p$ from $k$ to find the directrix

Step 8.2: Directrix is $y = \frac{9}{40} - \frac{9}{10}$

Step 9: Summarize the properties of the parabola

Direction: Opens Up

Vertex: $(-\frac{3}{10}, \frac{9}{40})$

Focus: $(-\frac{3}{10}, \frac{9}{8})$

Axis of Symmetry: $x = -\frac{3}{10}$

Directrix: $y = -\frac{27}{40}$

Step 10: Graph the parabola using the obtained properties

Knowledge Notes:

To solve for the center of a parabola, we need to convert the given equation into vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The process involves completing the square for the $x$-terms and isolating $y$. Once in vertex form, we can easily identify the vertex, which gives us the center of the parabola. Additionally, we can determine the direction the parabola opens (upwards if $a > 0$, downwards if $a < 0$), the focus, the axis of symmetry (which is the line $x = h$), and the directrix (a horizontal line at $y = k - p$ for an upward-opening parabola). The focus is a point where all the reflected rays from the parabola converge, and it is located at a distance $p = \frac{1}{4a}$ from the vertex along the axis of symmetry. The directrix is a fixed line that, together with the focus, defines the parabola as the set of all points equidistant from the focus and the directrix.