In Exercises 1–18, graph each ellipse and locate the foci. 25x²+4y² = 100

Verified step by step guidance

1. Step 1: Start by rewriting the given equation of the ellipse in standard form. The given equation is 25x^2 + 4y^2 = 100. Divide every term by 100 to simplify it.
2. Step 2: Simplify the equation to get \(\frac{x^2}{4} + \frac{y^2}{25} = 1\). This is the standard form of an ellipse equation where \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
3. Step 3: Identify the values of \(a^2\) and \(b^2\). Here, \(a^2 = 4\) and \(b^2 = 25\). Since \(b^2 > a^2\), this is a vertical ellipse.
4. Step 4: Calculate the values of \(a\) and \(b\). \(a = \sqrt{4} = 2\) and \(b = \sqrt{25} = 5\).
5. Step 5: Determine the foci of the ellipse. Use the formula \(c^2 = b^2 - a^2\) to find \(c\). Calculate \(c\) and then locate the foci at \((0, \pm c)\) since the ellipse is vertical.

Key Concepts

Ellipse Standard Form

An ellipse is defined by its standard form equation, which is typically written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis. Understanding this form is crucial for graphing the ellipse and identifying its key features, such as the foci and vertices.

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 denoted as c, where c² = a² - b². Knowing how to calculate and locate the foci helps in accurately graphing the ellipse.

Graphing Techniques

Graphing an ellipse involves plotting its center, vertices, and foci, and then sketching the curve that connects these points. Techniques include determining the lengths of the axes from the standard form, identifying the orientation of the ellipse (horizontal or vertical), and ensuring symmetry about the center. Mastery of these techniques is vital for creating an accurate representation of the ellipse.