Problem 8 - Algebra for College Students

Question

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 2 x^{2}+5 y^{2}=50 $$

Step-by-Step Solution

Answer

Vertices: (±5, 0); Minor Axis Endpoints: (0, ±√10); Foci: (±√15, 0).

Step 1: Identify the General Form of the Ellipse

The given equation is \(2x^2 + 5y^2 = 50\). The general form of the ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). To convert to this form, divide every term by 50 to simplify the equation.

Step 2: Simplify the Equation

Divide each term by 50: \(\frac{2x^2}{50} + \frac{5y^2}{50} = 1\), which simplifies to \(\frac{x^2}{25} + \frac{y^2}{10} = 1\). This is the equation of an ellipse centered at the origin \((0,0)\) with \(a^2 = 25\) and \(b^2 = 10\).

Step 3: Find the Vertices

For an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the vertices are \((\pm a, 0)\). Here, \(a = \sqrt{25} = 5\). Thus, the vertices are \((5, 0)\) and \((-5, 0)\).

Step 4: Find the Endpoints of the Minor Axis

The endpoints of the minor axis are \((0, \pm b)\). Here, \(b = \sqrt{10}\). So, the endpoints of the minor axis are \((0, \sqrt{10})\) and \((0, -\sqrt{10})\).

Step 5: Calculate the Foci of the Ellipse

The distance to the foci \(c\) is given by \(c = \sqrt{a^2 - b^2}\). Calculate \(c: c = \sqrt{25 - 10} = \sqrt{15}\). The foci are at \((\pm \sqrt{15}, 0)\).

Step 6: Sketch the Graph

Draw the ellipse with the center at the origin \((0, 0)\), vertices at \((5, 0)\) and \((-5, 0)\), endpoints of the minor axis at \((0, \sqrt{10})\) and \((0, -\sqrt{10})\), and locate the foci at \((\sqrt{15}, 0)\) and \((-\sqrt{15}, 0)\).

Key Concepts

Vertices of an Ellipse

In the world of ellipses, the vertices are very important as they represent the points where the ellipse reaches its longest extent along a specific axis. To find these, we start by identifying the standard form of the equation of an ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, the vertices are located along the "major axis", which is determined by the larger value between \(a\) and \(b\). The vertices are found at \((\pm a, 0)\) if the major axis is horizontal (meaning \(a > b\)), or at \((0, \pm b)\) if the major axis is vertical (meaning \(b > a\)).

• For the ellipse \( \frac{x^2}{25} + \frac{y^2}{10} = 1 \), we notice \(a^2 = 25\) and \(b^2 = 10\) meaning \(a = 5\) and \(b \approx 3.16\).
• This tells us our major axis is horizontal, thus the vertices are at \((5, 0)\) and \((-5, 0)\).

Knowing the position of the vertices helps us understand the orientation and placement of the ellipse. They are vital in visualizing how stretched the ellipse is in each direction.

Minor Axis of an Ellipse

The minor axis of an ellipse is the shortest axis, perpendicular to the major axis. If the ellipse's equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the endpoints of the minor axis can be found at \((0, \pm b)\) or \((\pm b, 0)\), depending on whether the major axis lies along the x-axis or y-axis.

• In our specific ellipse \( \frac{x^2}{25} + \frac{y^2}{10} = 1 \), \(b = \sqrt{10} \approx 3.16\).
• Given that \(a > b\), the minor axis is vertical, so its endpoints are at \((0, \sqrt{10})\) and \((0, -\sqrt{10})\).

Identifying the minor axis is critical for correctly sketching the shape of an ellipse and understanding its relative dimensions. It's the axis where the ellipse is least stretched.

Foci of an Ellipse

While the vertices and axes determine the shape and size of an ellipse, the foci impart significant information about its position and eccentricity. In an ellipse, there are two foci located symmetrically along the major axis. The distance from the center to each focus, denoted as \(c\), is calculated using \(c = \sqrt{a^2 - b^2}\).

• For our given ellipse, with \(a^2 = 25\) and \(b^2 = 10\), we find \(c = \sqrt{25 - 10} = \sqrt{15} \approx 3.87\).
• The foci are therefore located at \((\pm \sqrt{15}, 0)\), meaning \((\pm 3.87, 0)\).

Placement of the foci within the ellipse influences its shape; the closer they are to each other, the more circular the ellipse, while farther apart means a more elongated shape. Understanding the foci is key to insights about the ellipse’s geometry and overall structure.