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

Verified step by step guidance

1. Rewrite the given equation in the standard form of an ellipse by completing the square for both the x and y terms.
2. Group the x terms together and the y terms together: \(9x^2 - 18x + 4y^2 + 8y = 23\).
3. Complete the square for the x terms: Factor out the 9 from \(9x^2 - 18x\) to get \(9(x^2 - 2x)\). Add and subtract \(1\) inside the parentheses to complete the square: \(9((x-1)^2 - 1)\).
4. Complete the square for the y terms: Factor out the 4 from \(4y^2 + 8y\) to get \(4(y^2 + 2y)\). Add and subtract \(1\) inside the parentheses to complete the square: \(4((y+1)^2 - 1)\).
5. Rewrite the equation in standard form: \(\frac{(x-1)^2}{4} + \frac{(y+1)^2}{9} = 1\). Identify the center, vertices, and foci of the ellipse from this equation.

Key Concepts

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. Understanding the location of the foci is important for graphing the ellipse accurately and for applications involving the properties of ellipses.