In Exercises 1–18, graph each ellipse and locate the foci. x²/49 +y²/81 = 1

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1. Identify the standard form of the ellipse equation: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
2. Compare the given equation \( \frac{x^2}{49} + \frac{y^2}{81} = 1 \) with the standard form to determine \( a^2 = 49 \) and \( b^2 = 81 \).
3. Calculate \( a \) and \( b \) by taking the square roots: \( a = \sqrt{49} = 7 \) and \( b = \sqrt{81} = 9 \).
4. Since \( b > a \), the major axis is vertical. The vertices are at \((0, \pm b) = (0, \pm 9)\) and the co-vertices are at \((\pm a, 0) = (\pm 7, 0)\).
5. Find the foci using \( c^2 = b^2 - a^2 \). Calculate \( c = \sqrt{81 - 49} = \sqrt{32} \) and locate the foci at \((0, \pm c)\).

Key Concepts

Standard Form of an Ellipse

An ellipse in standard form is written as (x^2/a^2) + (y^2/b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying which denominator is larger helps determine the ellipse's orientation (horizontal or vertical). This form is essential for graphing the ellipse accurately.

Graphing an Ellipse

To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connecting these points smoothly forms the ellipse. Understanding the axes lengths and orientation is crucial for an accurate sketch.

Foci of an Ellipse

The foci are two fixed points inside the ellipse located along the major axis, found using c^2 = |a^2 - b^2|, where c is the distance from the center to each focus. Knowing how to calculate and plot the foci helps in understanding the ellipse's geometric properties.