Problem 45 - A Graphical Approach to College Algebra

Question

Find the eccentricity \(e\) of each ellipse or hyperbola. $$4 x^{2}+7 y^{2}=28$$

Step-by-Step Solution

Answer

The eccentricity is \(e = \sqrt{\frac{3}{7}}\).

Step 1: Identify the Conic Section

The given equation is \(4x^2 + 7y^2 = 28\). Since it has the form \(Ax^2 + By^2 = C\) where \(A\) and \(B\) are positive and unequal, it represents an ellipse.

Step 2: Standard Form of Ellipse Equation

Rearrange the equation to the standard form of an ellipse. Divide every term by 28 to obtain \(\frac{x^2}{7} + \frac{y^2}{4} = 1\). Here, \(a^2 = 7\) and \(b^2 = 4\).

Step 3: Identify Semi-Major and Semi-Minor Axis

Determine which of \(a^2\) or \(b^2\) is larger. Since \(a^2 = 7\) is greater than \(b^2 = 4\), \(a^2 = 7\) identifies the semi-major axis and \(b^2 = 4\) as the semi-minor axis.

Step 4: Calculate Eccentricity of Ellipse

For an ellipse, the eccentricity \(e\) is given by \(e = \sqrt{1 - \frac{b^2}{a^2}}\). Substitute \(a^2 = 7\) and \(b^2 = 4\) into the formula: \(e = \sqrt{1 - \frac{4}{7}}\).

Step 5: Simplify the Eccentricity Expression

Simplify \(e = \sqrt{1 - \frac{4}{7}} = \sqrt{\frac{7-4}{7}} = \sqrt{\frac{3}{7}}\).

Step 6: Finalize the Eccentricity

The eccentricity \(e\) of the given ellipse is \(e = \sqrt{\frac{3}{7}}\).

Key Concepts

Ellipse

An ellipse is a fascinating and important shape in geometry. It's a closed, symmetrical curve that resembles a stretched circle. Unlike a circle, which has one radius, an ellipse has two main lines you need to know about: the major axis and the minor axis. These axes give the ellipse its distinct shape. The longer axis is called the major axis, while the shorter one is the minor.

Ellipses have several unique properties. For one, the sum of the distances from any point on the ellipse to two fixed points called the foci remains constant. This characteristic forms the basis of its definition.

• An ellipse is often found in the orbit paths of planets.
• It's also seen in technical drawings and designs, essential in many fields such as astronomy and physics.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. Depending on the angle and position of the intersection, different shapes can emerge. The four main types of conic sections are ellipses, parabolas, hyperbolas, and circles. Each of these shapes has unique properties and equations.

In the case of an ellipse, the plane cuts the cone at an angle that is neither parallel to the base nor perpendicular to its axis. This specific angle reveals the elliptical curve.

• Conic sections are very important in the math world, especially in algebra and geometry.
• They model real-world phenomena, such as the trajectory of planets and projectiles.

Understanding conic sections is crucial for solving various geometric problems, including identifying the type of curve from its equation.

Standard Form of Ellipse

The standard form of an ellipse's equation helps us easily recognize its characteristics. This form is crucial for solving problems related to ellipses, such as finding the eccentricity or the lengths of the axes. The equation is written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2\) and \(b^2\) are the squares of the lengths of the semi-major and semi-minor axes respectively.

By rearranging and transforming the general equation into this standard form, we can quickly identify key properties:

• If \( a > b \), the major axis is along the x-direction.
• If \( a < b \), it lies along the y-direction.

The standard form allows us to understand the ellipse's orientation and size, making it indispensable for computations involving ellipses.

Semi-Major and Semi-Minor Axis

The semi-major and semi-minor axes are the principal axes of an ellipse, defining its shape and size. The semi-major axis is half the length of the longest diameter, while the semi-minor axis is half of the shortest. These axes are fundamental in determining how "stretched" the ellipse appears compared to a circle. In our exercise, analysis showed that \(a^2 = 7\) and \(b^2 = 4\), making the semi-major axis longer and dominating the ellipse's horizontal stretch.

• The semi-major axis always possesses the greater value.
• Using these axes, you can calculate the ellipse's eccentricity, a vital attribute showcasing its "flattening."

These parameters aren't just numbers; they reveal critical aspects of the ellipse's geometry, necessary for both theoretical and practical applications.