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Find the center and foci of the ellipse: $\newline$ $4x^{2}+9y^{2}+8x-36y+4=0$

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Write equations of ellipses in standard form using properties

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Q. Find the center and foci of the ellipse:

\newline

4x^{2}+9y^{2}+8x-36y+4=0

Complete the square: Complete the square for the $x$ -terms and $y$ -terms in the equation $4x^2 + 9y^2 + 8x - 36y + 4 = 0$ .
Group x and y terms: Group the x-terms and y-terms together: $(4x^2 + 8x) + (9y^2 - 36y) = -4$ .
Simplify coefficients: Divide the equation by $4$ to simplify the coefficients: $x^2 + 2x + \left(\frac{9}{4}\right)y^2 - 9y = -1$ .
Add squares for $x$ and $y$ : Complete the square for $x$ by adding $\left(\frac{2}{2}\right)^2 = 1$ to both sides and for $y$ by adding $\left(\frac{9}{2}\right)^2 / \left(\frac{9}{4}\right) = \frac{9}{4}$ to both sides: $\left(x^2 + 2x + 1\right) + \left(\frac{9}{4}\right)\left(y^2 - 4y + 4\right) = -1 + 1 + \frac{9}{4}$ .
Rewrite equation: Rewrite the equation as $(x + 1)^2 + \frac{9}{4}(y - 2)^2 = 9$ and then divide by $9$ to get the standard form of the ellipse: $\frac{(x + 1)^2}{9} + \frac{(y - 2)^2}{4} = 1$ .
Identify center and foci: Identify the center of the ellipse as $(-1, 2)$ and calculate the distance $c$ from the center to the foci using the formula $c^2 = a^2 - b^2$ , where $a^2 = 9$ and $b^2 = 4$ . Thus, $c^2 = 9 - 4 = 5$ , so $c = \sqrt{5}$ . The foci are located at $(-1, 2 + \sqrt{5})$ and $(-1, 2 - \sqrt{5})$ .

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