Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{1}{2}\), vertices on the \(x\)-axis, passing through \((1,3)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{4} + \frac{y^2}{3} = 1\).

Step 1: Understanding the Problem

The ellipse is centered at the origin and has a given eccentricity of \(\frac{1}{2}\). Its vertices are along the \(x\)-axis, which tells us that the major axis is horizontal. We are given a point \((1,3)\) on the ellipse.

Step 2: Applying Eccentricity Formula

For an ellipse with center at the origin and major axis along \(x\)-axis, the equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(c^2 = a^2 - b^2\). Here, the eccentricity \(e\) is \(\frac{c}{a}\). With \(e = \frac{1}{2}\), we have \(c = \frac{a}{2}\).

Step 3: Substituting Eccentricity

Substituting \(c = \frac{a}{2}\) into \(c^2 = a^2 - b^2\), gives \(\left(\frac{a}{2}\right)^2 = a^2 - b^2\). Simplify to get \(\frac{a^2}{4} = a^2 - b^2\), or \(b^2 = \frac{3}{4}a^2\).

Step 4: Using Given Point

Substitute \(x = 1\) and \(y = 3\) into the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This gives \(\frac{1}{a^2} + \frac{9}{b^2} = 1\). Substitute \(b^2 = \frac{3}{4}a^2\) into the equation.

Step 5: Solving for \(a^2\)

From \(\frac{1}{a^2} + \frac{9}{\frac{3}{4}a^2} = 1\), simplify to \(\frac{1}{a^2} + \frac{12}{4a^2} = 1\) or \(\frac{1}{a^2} + \frac{3}{a^2} = 1\). This results in \(\frac{4}{a^2} = 1\), hence \(a^2 = 4\).

Step 6: Finding \(b^2\)

Using \(b^2 = \frac{3}{4}a^2\) and \(a^2 = 4\), substitute to get \(b^2 = \frac{3}{4} \times 4 = 3\).

Step 7: Conclusion

Now we have \(a^2 = 4\) and \(b^2 = 3\). The equation of the ellipse is \(\frac{x^2}{4} + \frac{y^2}{3} = 1\).

Key Concepts

Eccentricity

Eccentricity is a measure that describes how stretched out an ellipse appears. It tells us about the shape of the ellipse. In mathematical terms, eccentricity (\(e\)) is the ratio of the distance from the center of the ellipse to its focus (\(c\)) compared to the distance from the center to a vertex on the major axis (\(a\)).
For ellipses:

• The eccentricity (\(e\)) is always between 0 and 1, where 0 is a perfect circle.
• In this exercise, given eccentricity is \(\frac{1}{2}\).
• We use the formula \(e = \frac{c}{a}\) to find the relationship between these distances.

Knowing \(e = \frac{1}{2}\) gives us \(c = \frac{a}{2}\), indicating the focus is halfway from the center to the vertex, along the major axis. This impacts how the ellipse looks and helps determine its equation.

Ellipse Centered at Origin

Placing an ellipse centered at the origin simplifies its equation. The center of an ellipse is the midpoint of both its major and minor axes.
At the origin, the general equation of an ellipse is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here:

• \(a\) refers to the semi-major axis length.
• \(b\) is the semi-minor axis length.

This equation assumes a simple symmetrical spread across both axes, with no tilting. It makes calculations more straightforward for determining points and properties of the ellipse.Since the ellipse is centered at the origin, locating any point on it, such as \((1, 3)\), becomes easier by substituting the values into this standard form equation.

Vertices on x-axis

When the vertices of an ellipse lie on the x-axis, it implies that the major axis runs horizontally. This orientation affects the standard form of the ellipse equation because:

• The major axis is tied with \(x\), meaning \(a\) belongs in the \(x^2\) denominator.
• The vertex positions directly affect values of \(a\) since they are located at \((-a, 0)\) and \((a, 0)\).

Making connections to the eccentricity, for this problem, we express it as \(\frac{x^2}{4} + \frac{y^2}{3} = 1\), revealing the scope of the x-axis as the broader reach (major axis).
Ultimately, when working out any point through such an ellipse, this information reflects the breadth and orientation essential in forming its graph and understanding its physical arrangement in space.