In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1

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1. Identify the standard form of the ellipse equation: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
2. Compare the given equation \(\frac{(x+1)^2}{36} + \frac{(y-4)^2}{4} = 1\) with the standard form to determine the values of \(h\), \(k\), \(a^2\), and \(b^2\).
3. Determine the center of the ellipse \((h, k)\) which is \((-1, 4)\).
4. Identify the lengths of the semi-major axis \(a\) and the semi-minor axis \(b\) by taking the square roots of \(a^2\) and \(b^2\), respectively. Here, \(a = 6\) and \(b = 2\).
5. Calculate the distance of the foci from the center using the formula \(c = \sqrt{a^2 - b^2}\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse is given by the equation (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis. This form allows us to easily identify the center and the lengths of the axes, which are crucial for graphing the ellipse.

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 calculated using the formula c = √(a² - b²), where c is the distance to the foci, a is the semi-major axis, and b is the semi-minor axis. Understanding the location of the foci helps in accurately graphing the ellipse.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, determining the lengths of the semi-major and semi-minor axes, and marking the foci. The axes are drawn perpendicular to each other, and the ellipse is sketched by connecting points that satisfy the ellipse's equation. This visual representation is crucial for understanding the properties and dimensions of the ellipse.