Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23 = 0

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Rewrite the given equation in standard form by grouping the x-terms and y-terms together: \( 9x^2 - 18x + 4y^2 + 8y = 23 \).

Complete the square for the x-terms. Factor out the coefficient of \(x^2\) (which is 9) from \(9x^2 - 18x\): \(9(x^2 - 2x)\). Then, complete the square inside the parentheses by adding and subtracting \((\frac{-2}{2})^2 = 1\): \(9(x^2 - 2x + 1 - 1) = 9((x - 1)^2 - 1)\).

Complete the square for the y-terms. Factor out the coefficient of \(y^2\) (which is 4) from \(4y^2 + 8y\): \(4(y^2 + 2y)\). Then, complete the square inside the parentheses by adding and subtracting \((\frac{2}{2})^2 = 1\): \(4(y^2 + 2y + 1 - 1) = 4((y + 1)^2 - 1)\).

Substitute the completed squares back into the equation: \(9((x - 1)^2 - 1) + 4((y + 1)^2 - 1) = 23\). Simplify the constants: \(9(x - 1)^2 - 9 + 4(y + 1)^2 - 4 = 23\), which simplifies further to \(9(x - 1)^2 + 4(y + 1)^2 = 36\).

Divide through by 36 to express the equation in standard form: \(\frac{(x - 1)^2}{4} + \frac{(y + 1)^2}{9} = 1\). From this, identify the center \((1, -1)\), the semi-major axis length \(3\), the semi-minor axis length \(2\), and calculate the foci using \(c = \sqrt{a^2 - b^2}\), where \(a = 3\) and \(b = 2\).

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주요 개념

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Ellipse Standard Form

An ellipse is defined by its standard form equation, which typically looks like (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis. To graph an ellipse, it is essential to convert the given equation into this standard form, allowing for easy identification of its key features.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique is crucial for rewriting the ellipse equation in standard form, as it helps isolate the variables and identify the center and axes of the ellipse. It involves manipulating the equation to create a squared term for both x and y.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis, which are essential for defining the shape of the ellipse. The distance from the center to each focus is denoted as c, where c² = a² - b². Identifying the foci is important for understanding the ellipse's properties, such as its eccentricity and how it relates to the distance from any point on the ellipse to the foci.