In Exercises 1–18, graph each ellipse and locate the foci. x²/16+y²/4 = 1

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1. Identify the standard form of the ellipse equation: $\frac{x^2}{16}+\frac{y^2}{4}=1$. Here, $a^2\; =\; 16$ and $b^2\; =\; 4$.
2. Determine the values of $a$ and $b$ by taking the square roots: and $b\; =\; 2$. Since $a^2\; >\; b^2$, the major axis is along the x-axis.
3. Calculate the focal distance $c$ using the relationship $c^2\; =\; a^2\; -\; b^2$. Substitute the values to get $c^2\; =\; 16\; -\; 4$.
4. Find the coordinates of the foci, which lie along the major axis at $(�c,\; 0)$ and $(c,\; 0)$.
5. Sketch the ellipse centered at the origin with vertices at and $(-a,\; 0)$, co-vertices at $(0,\; b)$ and $(0,\; -b)$, and mark the foci at the points found in the previous step.

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 a and b helps determine the shape and orientation of the ellipse on the coordinate plane.

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, showing its size and orientation.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse, found using c^2 = a^2 - b^2, where c is the distance from the center to each focus. Knowing the foci is essential for understanding ellipse properties and its geometric definition.