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Animated Solution for Mathematics - Conic Sections: If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Q: Why is the distance between foci $2ae$?

Think of the foci as being located at $(ae, 0)$ and $(-ae, 0)$. The distance is the difference in their x-coordinates: $ae - (-ae) = 2ae$. It is a standard result, but deriving it from the coordinates ensures you never forget it.

Select Answer:

Visualized Solution

Standard Ellipse Equation

Let the standard ellipse be $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$
Assume $a > b$ , so the major axis is along the x-axis.

Length of Minor Axis

The minor axis lies along the y-axis.
Total length of the minor axis is $2 b$ .

Distance Between Foci

The foci $S$ and $S^{'}$ lie on the major axis.
Coordinates are $(a e, 0)$ and $(- a e, 0)$ .
Distance between foci is $2 a e$ .

Applying the Given Condition

According to the problem:
Minor Axis = $\frac{1}{2} \times$ (Distance between foci)

Substituting the Lengths

Substitute the known expressions:
$2 b = \frac{1}{2} (2 a e)$

Simplifying the Equation

Cancel the $2$ on the right side:
$2 b = a e$
Rearrange to find the ratio $\frac{b}{a}$ :
$\frac{b}{a} = \frac{e}{2}$

Standard Eccentricity Relation

For an ellipse, the eccentricity $e$ is given by:
$e^{2} = 1 - \frac{b ^{2}}{a ^{2}}$
Or, $e^{2} = 1 - (\frac{b}{a})^{2}$

Substituting $\frac{b}{a}$

Substitute $\frac{b}{a} = \frac{e}{2}$ into the eccentricity formula:
$e^{2} = 1 - (\frac{e}{2})^{2}$

Expanding the Square

Square the term inside the bracket:
$e^{2} = 1 - \frac{e ^{2}}{4}$

Grouping $e^{2}$ Terms

Move $- \frac{e ^{2}}{4}$ to the left side:
$e^{2} + \frac{e ^{2}}{4} = 1$

Solving for $e^{2}$

Take the common denominator on the left:
$\frac{4 e ^{2} + e ^{2}}{4} = 1$
$\frac{5 e ^{2}}{4} = 1$
$e^{2} = \frac{4}{5}$

Final Eccentricity

Take the square root on both sides:
$e = \frac{4}{5}$
$e = \frac{2}{5}$
(Since eccentricity $e > 0$ )

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The Sigma Insight: Foci, Directrices, and Eccentricity

Solution Diagram

Analyzing the Setup

The geometry of the ellipse is defined by its semi-major axis

a

, semi-minor axis

b

, and eccentricity

e

. We begin with the standard equation of an ellipse centered at the origin:

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

Assuming

a > b

, the ellipse is stretched horizontally.

Defining the Players

The minor axis spans from

(0, - b)

(0, b)

, giving it a total length of

2 b

. The foci are located at

(\pm a e, 0)

, meaning the distance between the two foci is

2 a e

These two values,

2 b

and

2 a e

, are the primary components of our geometric constraint.

The Bridge

The problem states that the length of the minor axis is equal to half the distance between the foci. Mathematically, this is expressed as:

2 b = \frac{1}{2} (2 a e)

Simplifying this, we obtain

2 b = a e

. Rearranging for the ratio of the axes, we find:

\frac{b}{a} = \frac{e}{2}

The Algebraic Triumph

We now invoke the fundamental identity relating eccentricity to the semi-axes:

e^{2} = 1 - \frac{b ^{2}}{a ^{2}} = 1 - (\frac{b}{a})^{2}

Substituting our ratio

\frac{b}{a} = \frac{e}{2}

into this identity, we get:

e^{2} = 1 - (\frac{e}{2})^{2}

Expanding the expression yields:

e^{2} = 1 - \frac{e ^{2}}{4}

Grouping the

e^{2}

terms on one side:

e^{2} + \frac{e ^{2}}{4} = 1 \Rightarrow \frac{5 e ^{2}}{4} = 1

Solving for

e^{2}

, we find

e^{2} = \frac{4}{5}

. Taking the square root, we arrive at the final result:

$e = \frac{2}{5}$

Common Traps & FAQs

Wait, why is the minor axis

2 b

? I thought it was just

b

Great question! The minor axis is the *entire* segment connecting the two endpoints on the y-axis. Since the center is at

(0, 0)

and the endpoints are

(0, b)

and

(0, - b)

, the total length is

b - (- b) = 2 b

. Always look at the full diameter!

Why is the distance between foci

2 a e

Think of the foci as being located at

(a e, 0)

and

(- a e, 0)

. The distance is the difference in their x-coordinates:

a e - (- a e) = 2 a e

. It is a standard result, but deriving it from the coordinates ensures you never forget it.

Can the eccentricity

e

ever be negative?

No! Eccentricity is a measure of 'flatness' or deviation from a circle. By definition,

e

is the ratio of the distance from the center to a focus (

a e

) to the distance from the center to a vertex (

a

). Since both are lengths,

e

must be positive. For an ellipse,

0 < e < 1

Animated Solution for Mathematics - Conic Sections: If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Select Answer:

Visualized Solution

Standard Ellipse Equation

Length of Minor Axis

Distance Between Foci

Applying the Given Condition

Substituting the Lengths

Simplifying the Equation

Standard Eccentricity Relation

Substituting $\frac{b}{a}$

Expanding the Square

Grouping $e^{2}$ Terms

Solving for $e^{2}$

Final Eccentricity

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

Defining the Players

The Bridge

The Algebraic Triumph

Similar Questions

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Animated Solution for Mathematics - Conic Sections: If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Select Answer:

Visualized Solution

Standard Ellipse Equation

Length of Minor Axis

Distance Between Foci

Applying the Given Condition

Substituting the Lengths

Simplifying the Equation

Standard Eccentricity Relation

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ab​

Expanding the Square

Grouping .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } e2 Terms

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } e2

Final Eccentricity

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

Defining the Players

The Bridge

The Algebraic Triumph

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