Questions: Find the foci of the ellipse. x^2/5 + y^2/21 = 1 (0,16) and (0,-16) (16,0) and (-16,0) (0,4) and (0,-4) (4,0) and (-4,0)

Transcript text: Find the foci of the ellipse. \[ \frac{x^{2}}{5}+\frac{y^{2}}{21}=1 \] $(0,16)$ and $(0,-16)$ $(16,0)$ and $(-16,0)$ $(0,4)$ and $(0,-4)$ $(4,0)$ and $(-4,0)$

Solution

Solution Steps

To find the foci of the ellipse given by the equation $\frac{x^{2}}{5}+\frac{y^{2}}{21}=1$, we first identify the semi-major and semi-minor axes. The larger denominator corresponds to the semi-major axis, which is along the y-axis in this case. We calculate the distance of the foci from the center using the formula $c = \sqrt{b^2 - a^2}$, where $b$ is the semi-major axis and $a$ is the semi-minor axis.

Step 1: Identify the Semi-Major and Semi-Minor Axes

The given ellipse is represented by the equation

\[ \frac{x^{2}}{5} + \frac{y^{2}}{21} = 1. \]

From this equation, we can identify the semi-minor axis $a$ and the semi-major axis $b$ as follows:

\[ a^{2} = 5 \quad \text{and} \quad b^{2} = 21. \]

Step 2: Calculate the Distance to the Foci

To find the distance $c$ from the center of the ellipse to each focus, we use the formula

\[ c = \sqrt{b^{2} - a^{2}}. \]

Substituting the values we have:

\[ c = \sqrt{21 - 5} = \sqrt{16} = 4. \]

Step 3: Determine the Coordinates of the Foci

Since the semi-major axis is along the y-axis, the coordinates of the foci are given by