91Ó°ÊÓ

In geometry, ellipses are defined by specific equations that describe their shape. The standard form for the equation of an ellipse is: \ \ \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \ \ Here, \(a\) represents the semi-major axis and \(b\) represents the semi-minor axis. \ \ - If \(a > b\), the ellipse is stretched along the x-axis.
- If \(b > a\), the ellipse is stretched along the y-axis.
- The values \(a\) and \(b\) are derived from the coefficients in the denominator of \(x^{2}\) and \(y^{2}\). For instance, in the given equation \( \frac{x^{2}}{25}+\frac{y^{2}}{4}=1 \), we see that \( a^{2} = 25 \) and \( b^{2} = 4 \). \ \ This implies \( a = 5 \) and \( b = 2 \). Hence, the ellipse stretches more along the x-axis.

The vertices of an ellipse are the points where the ellipse intersects its major axis. Knowing how to find these points is crucial for understanding the ellipse's orientation on the graph. \ \ For an ellipse centered at the origin, the vertices are located at \((\text{center} \ \ \pm a, 0)\) for a horizontal ellipse, and \((0, \ \ \text{center} \pm b)\) for a vertical ellipse. \ \ Given \( \frac{x^{2}}{25}+\frac{y^{2}}{4}=1 \), the center is at \((0,0)\). \ \ Plugging in our values, the vertices are: \ \ - \((\textbf{+5, 0})\) \ - \((-5, 0)\). These points represent the farthest extent of the ellipse along the x-axis.

The foci (singular: focus) of an ellipse are two special points such that the sum of the distances from any point on the ellipse to the foci is constant. \ \ To find the foci, we use the formula \( c = \sqrt{a^{2} - b^{2}} \), where \(c\) is the distance from the center to each focus. \ \ For our equation \( c = \sqrt{25 - 4} = \sqrt{21} \approx 4.58 \) \ \ The foci are thus located at: \ \ - \((\text{center} \ \ \pm c, 0)\) \ Thus, our foci for \( \frac{x^{2}}{25}+\frac{y^{2}}{4}=1\) are: \ \ - \((4.58, 0)\)
- \((-4.58, 0)\).

Graphing an ellipse involves plotting the center, vertices, and foci, then sketching the shape accurately. \ \ Follow these steps: \ \ - Start by plotting the center of the ellipse at \((0, 0)\). \ \ - Mark the vertices at \((\textbf{+5, 0})\) and \((-5, 0)\). \ - Next, mark the foci at \((\textbf{+4.58, 0})\) and \((-4.58, 0)\). \ \ Draw a smooth, oval-like curve passing through the vertices, ensuring to maintain a symmetric shape around the center. \ \ The curve should enclose the foci, making the distances from any point on the ellipse to the two foci constant. \ \ Ensure your graph captures the ellipse's elongated nature along the x-axis, reflecting the greater stretch represented by the vertices.