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

Q: Why do we square the equation $b = \frac{ae}{4}$?

We square it because our fundamental relation, $b^2 = a^2(1 - e^2)$, involves $b^2$. By squaring our derived condition, we get $b^2$ on one side, which allows for a direct substitution. It is a standard algebraic technique to eliminate the square root and align the variables with the known formula.

Select Answer:

Visualized Solution

Standard Ellipse Equation

Let the equation of the ellipse be $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$
Assume $a > b$

Length of Minor Axis

The minor axis is the vertical segment.
Length of Minor Axis = $2 b$

Distance Between Foci

The foci are located at $(\pm a e, 0)$ .
Distance between Foci = $2 a e$

Applying the Given Condition

According to the question:
Length of Minor Axis = $\frac{1}{4} \times$ (Distance between Foci)

Setting Up the Equation

Substitute the geometric lengths:
$2 b = \frac{1}{4} (2 a e)$

Simplifying the Equation

Divide both sides by $2$ :
$b = \frac{a e}{4}$

Squaring Both Sides

Square both sides to eliminate fractions and prepare for substitution:
$b^{2} = (\frac{a e}{4})^{2}$
$b^{2} = \frac{a ^{2} e ^{2}}{16}$

The Fundamental Relation

Recall the standard eccentricity formula for an ellipse:
$b^{2} = a^{2} (1 - e^{2})$

Substituting $b^{2}$

Substitute $b^{2} = a^{2} (1 - e^{2})$ into our previous equation:
$a^{2} (1 - e^{2}) = \frac{a ^{2} e ^{2}}{16}$

Canceling $a^{2}$

Since $a \neq = 0$ , divide both sides by $a^{2}$ :
$1 - e^{2} = \frac{e ^{2}}{16}$

Grouping $e^{2}$ Terms

Move $- e^{2}$ to the right side:
$1 = e^{2} + \frac{e ^{2}}{16}$

Combining the Terms

Take $e^{2}$ common on the right side:
$1 = e^{2} (1 + \frac{1}{16})$
$1 = e^{2} (\frac{17}{16})$

Isolating $e^{2}$

Multiply both sides by $\frac{16}{17}$ :
$e^{2} = \frac{16}{17}$

Final Calculation for Eccentricity

Take the square root of both sides:
$e = \frac{16}{17}$
$e = \frac{4}{17}$

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The Sigma Insight: Foci, Directrices, and Eccentricity

Solution Diagram

Analyzing the Setup

We define the ellipse using the standard equation:

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

The length of the minor axis is given by

2 b

, and the distance between the two foci located at

(\pm a e, 0)

2 a e

According to the problem, the length of the minor axis is one-fourth of the distance between the foci. We express this as:

2 b = \frac{1}{4} (2 a e)

By simplifying this expression, we obtain the relationship:

b = \frac{a e}{4}

The Bridge

Connecting the Variables

To find the eccentricity

e

, we utilize the fundamental property of an ellipse that relates the semi-axes to the eccentricity:

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

We square our previous relation,

b = \frac{a e}{4}

, to align it with this fundamental equation:

b^{2} = \frac{a ^{2} e ^{2}}{16}

Now, we substitute this expression for

b^{2}

into the fundamental relation:

a^{2} (1 - e^{2}) = \frac{a ^{2} e ^{2}}{16}

The Final Act

Solving for Eccentricity

Since $a eq 0$, we divide both sides by

a^{2}

to simplify the equation:

1 - e^{2} = \frac{e ^{2}}{16}

Rearranging the terms to isolate

e^{2}

, we add

e^{2}

to both sides:

1 = e^{2} + \frac{e ^{2}}{16}

1 = e^{2} (1 + \frac{1}{16}) = e^{2} (\frac{17}{16})

Solving for

e^{2}

, we find:

e^{2} = \frac{16}{17}

Taking the square root of both sides, we arrive at the final value for the eccentricity:

$e = \frac{4}{17}$

Common Traps & FAQs

Wait, why is the minor axis

2 b

and not just

b

That is a great question! In the standard equation of an ellipse,

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

, the length of the semi-minor axis is

b

. The full minor axis spans from the top vertex to the bottom vertex, which is

b - (- b) = 2 b

. Always remember to distinguish between the semi-axis and the full axis length!

Why can't we just use energy conservation or some other physics principle here?

This is a pure coordinate geometry problem, not a mechanics problem. While ellipses appear in planetary motion (Kepler's laws), here we are dealing with the static definition of the conic section. We rely on the geometric properties of the ellipse, specifically the relationship between the axes and the foci, to find the eccentricity.

Why do we square the equation

b = \frac{a e}{4}

We square it because our fundamental relation,

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

, involves

b^{2}

. By squaring our derived condition, we get

b^{2}

on one side, which allows for a direct substitution. It is a standard algebraic technique to eliminate the square root and align the variables with the known formula.

Animated Solution for Mathematics - Conic Sections: If the length of the minor axis of an ellipse is equal to one fourth 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

Setting Up the Equation

Simplifying the Equation

Squaring Both Sides

The Fundamental Relation

Substituting $b^{2}$

Canceling $a^{2}$

Grouping $e^{2}$ Terms

Combining the Terms

Isolating $e^{2}$

Final Calculation for Eccentricity

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

The Bridge

The Final Act

Similar Questions

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Animated Solution for Mathematics - Conic Sections: If the length of the minor axis of an ellipse is equal to one fourth 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

Setting Up the Equation

Simplifying the Equation

Squaring Both Sides

The Fundamental Relation

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b2

Canceling .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2

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

Combining the Terms

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

Final Calculation for Eccentricity

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

The Bridge

The Final Act

Similar Questions

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