/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 184 In an ellipse, the distance betw... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 184

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is [2006] (a) \(\frac{3}{5}\) (b) \(\frac{1}{2}\) (c) \(\frac{4}{5}\) (d) \(\frac{1}{\sqrt{5}}\)

Short Answer

Expert verified
The eccentricity is \(\frac{3}{5}\), which is option (a).

Step by step solution

01

Understand properties of an ellipse

An ellipse with its center at the origin has the general equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The distance between the foci is \(2c\), where \(c = \sqrt{a^2 - b^2}\). The eccentricity \(e\) is given by \(e = \frac{c}{a}\).
02

Calculate the semi-minor axis, b

We are given that the minor axis is 8, so \(b = \frac{8}{2} = 4\).
03

Calculate distance between foci, 2c

We know the distance between the foci is 6, so \(2c = 6\). This implies that \(c = \frac{6}{2} = 3\).
04

Calculate value of a using c and b

Using the formula \(c = \sqrt{a^2 - b^2}\), substitute \(c = 3\) and \(b = 4\): \(3 = \sqrt{a^2 - 4^2}\).
05

Solve for a

Square both sides to get \(9 = a^2 - 16\). Then solve for \(a^2\) by adding 16 to both sides: \(a^2 = 25\). Calculate \(a\): \(a = \sqrt{25} = 5\).
06

Calculate eccentricity e

The eccentricity \(e\) is calculated as \(e = \frac{c}{a} = \frac{3}{5}\).
07

Validate answer with given options

The answer \(e = \frac{3}{5}\) corresponds to option (a).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eccentricity
Eccentricity is a measure of how much an ellipse deviates from being a perfect circle. A circle has an eccentricity of 0, while an ellipse has a value that is greater than 0 but less than 1.
Eccentricity is denoted by the letter \( e \) and can be calculated using the formula:
  • \( e = \frac{c}{a} \)
where \( c \) is the distance from the center to a focus, and \( a \) is the semi-major axis of the ellipse.
A smaller eccentricity means the ellipse is more circular, while a larger eccentricity suggests it is more elongated. In our problem, where we found \( e = \frac{3}{5} \), the ellipse is moderately elongated.
Semi-Major Axis
The semi-major axis is the longest radius of an ellipse, extending from the center to the farthest edge. It is the distance denoted by \( a \). When thinking about an ellipse, imagine stretching a circle in one direction.
This stretch in the longer direction is described by the semi-major axis.
The formula that relates the semi-major axis to other parts of the ellipse is:
  • \( c = \sqrt{a^2 - b^2} \)
where \( b \) is the semi-minor axis.
In our given exercise, solving for \( a \) using the equation \( 3 = \sqrt{a^2 - 4^2} \) provides us with \( a = 5 \). This means the semi-major axis measures 5 units.
Semi-Minor Axis
The semi-minor axis is the shortest radius of the ellipse, stretching from its center to the nearest edge. This measurement is denoted by \( b \). It is crucial in defining the shape of the ellipse.
If the semi-minor axis is much shorter than the semi-major axis, the ellipse will appear more stretched. The relationship among the semi-major axis, semi-minor axis, and the foci can be expressed as:
  • \( b = \sqrt{a^2 - c^2} \)
The given problem mentions that the minor axis is 8. Therefore, the semi-minor axis \( b \) is \( \frac{8}{2} = 4 \). This means our ellipse is somewhat elongated.
Foci
The foci of an ellipse are two special points located along its major axis that play a key role in defining its shape. Any point on the ellipse has the same total distance to these two foci.
This is a defining property of ellipses. In formulas, when we say the distance between the foci is \( 2c \), \( c \) represents the distance from the center to one focus.
In our exercise, it is given that the distance between the foci is 6, hence \( 2c = 6 \), leading to \( c = 3 \).
These foci are vital in determining the eccentricity and hence the overall form of the ellipse.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The equation of a tangent to the parabola, \(x^{2}=8 y\), which makes an angle \(\theta\) with the positive direction of \(x\)-axis, is: (a) \(y=x \tan \theta+2 \cot \theta\) (b) \(y=x \tan \theta-2 \cot \theta\) (c) \(x=y \cot \theta+2 \tan \theta\) (d) \(x=y \cot \theta-2 \tan \theta\)

If the two circles \((x-1)^{2}+(y-3)^{2}=r^{2}\) and \(x^{2}+y^{2}-8 x+2 y+8=0\) intersect in two distinct point, then (a) \(r>2\) (b) \(2

If the chord \(y=m x+1\) of the circle \(x^{2}+y^{2}=1\) subtends an angle of measure \(45^{\circ}\) at the major segment of the circle then value of \(m\) is (a) \(2 \pm \sqrt{2}\) (b) \(-2 \pm \sqrt{2}\) (c) \(-1 \pm \sqrt{2}\) (d) none of these

The equation \(\operatorname{Im}\left(\frac{i z-2}{z-i}\right)+1=0, z \in C, z \neq i\) represents a part of a circle having radius equal to : (a) 2 (b) 1 (c) \(\frac{3}{4}\) (d) \(\frac{1}{2}\)

If the circles \(x^{2}+y^{2}-16 x-20 y+164=r^{2}\) and \((x-4)^{2}+(y-7)^{2}=36\) intersect at two distinct points, then: (a) \(\mathrm{r}>11\) (b) \(0<\mathrm{r}<1\) (c) \(\mathrm{r}=11\) (d) \(1
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.