Problem 13 - Contemporary Precalculus

Question

Find the eccentricity of the conic whose equation is given. $$\frac{x^{2}}{100}+\frac{y^{2}}{99}=1$$

Step-by-Step Solution

Answer

Answer: The eccentricity of the conic is $$\frac{1}{10}.$$

Step 1: Identify the values of a and b

From the equation, we can see that $$a^2 = 100 \text{ and } b^2 = 99.$$ We need these values to calculate the eccentricity, so let's take the square root of both of them to get: $$a = 10 \text{ and } b = 3\sqrt{11}.$$

Step 2: Calculate the eccentricity

Now we can use the eccentricity formula to find the eccentricity of the conic: $$e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{(3\sqrt{11})^2}{10^2}}$$ Calculating the eccentricity, $$e = \sqrt{1 - \frac{99}{100}} = \sqrt{\frac{1}{100}}$$ Thus, the eccentricity of the conic is $$e = \frac{1}{10}.$$

Key Concepts

Eccentricity

Eccentricity is a crucial concept in understanding conic sections. It helps us determine the shape of the conic. The eccentricity of a conic section, denoted as \(e\), tells us how "stretched" or "curved" the conic is.

For different conic sections, the eccentricity varies as follows:

• If \(e = 0\), it's a circle since there's no deviation from being perfectly round.
• For \(0 < e < 1\), the conic is an ellipse. The closer \(e\) is to 0, the more circular the ellipse.
• If \(e = 1\), it's a parabola. This happens when the ellipse is stretched to an extreme.
• If \(e > 1\), the conic is a hyperbola.

In the given exercise, we calculated \(e = \frac{1}{10}\), which is less than 1. Thus, the conic is an ellipse, being quite close to a circle.

Ellipse

An ellipse is a type of conic section that resembles an elongated circle. It's defined as the set of all points for which the sum of the distances to two fixed points (foci) is constant.

In the equation given, \(\frac{x^{2}}{100}+\frac{y^{2}}{99}=1\), we see an ellipse, as both denominators \(a^2\) and \(b^2\) are positive and different.
Here's what makes an ellipse special:

• The major axis is the longer diameter, and minor is the shorter. In this example, the minor axis corresponds to \(b^2 = 99\).
• The center of the ellipse is at the origin, \((0, 0)\), given no linear terms like \(x\) or \(y\) appear in the equation.

An ellipse is very versatile, appearing in celestial orbits, architectural structures, and more.

Equation of a conic

The equation of a conic is a powerful way to describe various shapes, including ellipses, parabolas, and hyperbolas. A conic section's equation typically appears in the form: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]for ellipses. This format helps us quickly identify the type of conic.

For the given exercise's equation, \(\frac{x^{2}}{100}+\frac{y^{2}}{99}=1\), it is essential to note:

• \(a^2 = 100\) and \(b^2 = 99\), leading to \(a = 10\) and \(b = 3\sqrt{11} \).
• The form of the equation directly indicates it's an ellipse due to the addition and positive values.

Understanding the equation allows us to define the conic's features and calculate properties like eccentricity. Conics are fundamental in various fields, including physics and engineering, where they describe orbits and trajectories.