Question

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:

• A. \( \frac{4}{\sqrt{17}} \)
• B. \( \frac{\sqrt{5}}{16} \)
• C. \( \frac{3}{\sqrt{19}} \)
• D. \( \frac{\sqrt{5}}{7} \)

Hint:

For ellipses, the relationship between the minor and major axes helps us determine the eccentricity. Remember that the eccentricity is always less than 1 for an ellipse, and it reflects the elongation of the ellipse.

Answer 1

The Correct Option is A

We are given that the length of the minor axis is equal to one fourth of the distance between the foci. Let the length of the minor axis be \( b \), and the length of the major axis be \( a \). The distance between the foci is \( 2c \), where \( c \) is the distance from the center to the foci. Given: \[ b = \frac{1}{4} \times 2c = \frac{c}{2} \] Now, the relationship between \( a \), \( b \), and \( c \) in an ellipse is: \[ c^2 = a^2 - b^2 \] Substitute \( b = \frac{c}{2} \) into the equation: \[ c^2 = a^2 - \left(\frac{c}{2}\right)^2 \] \[ c^2 = a^2 - \frac{c^2}{4} \] Multiplying both sides by 4 to eliminate the fraction: \[ 4c^2 = 4a^2 - c^2 \] \[ 5c^2 = 4a^2 \] \[ c^2 = \frac{4a^2}{5} \] Now, the eccentricity \( e \) of an ellipse is defined as: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \] Substitute the value of \( b \) and \( c \) into the equation: \[ e = \sqrt{1 - \frac{\left(\frac{c}{2}\right)^2}{a^2}} = \sqrt{1 - \frac{c^2}{4a^2}} \] Substitute \( c^2 = \frac{4a^2}{5} \) into the equation: \[ e = \sqrt{1 - \frac{\frac{4a^2}{5}}{4a^2}} = \sqrt{1 - \frac{1}{5}} \] \[ e = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \] Thus, the eccentricity is \( e = \frac{4}{\sqrt{17}} \).

Answer 2

In this problem, we are given a condition relating the length of the minor axis of an ellipse to the distance between its foci. We need to determine the eccentricity of this ellipse.

Concept Used:

For a standard ellipse with the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( a > b \)), the key properties are:

• Length of the minor axis = \( 2b \)
• Distance between the foci = \( 2ae \), where \( e \) is the eccentricity.
• The relationship between the semi-major axis (\(a\)), semi-minor axis (\(b\)), and eccentricity (\(e\)) is given by: \[ b^2 = a^2 (1 - e^2) \]

Step-by-Step Solution:

Step 1: Translate the given information into a mathematical equation using the standard parameters of an ellipse.

The length of the minor axis is \( 2b \).

The distance between the foci is \( 2ae \).

The problem states that the length of the minor axis is one fourth of the distance between the foci. Therefore, we can write:

\[ 2b = \frac{1}{4} (2ae) \]

Step 2: Simplify the equation derived in the previous step.

\[ 2b = \frac{ae}{2} \]

Multiplying both sides by 2, we get a simpler relation between \(b\), \(a\), and \(e\):

\[ 4b = ae \]

Step 3: Use the standard eccentricity formula to solve for \( e \).

The standard relationship is \( b^2 = a^2 (1 - e^2) \). We can substitute the expression for \( b \) from Step 2 into this formula. First, let's express \(b\) in terms of \(a\) and \(e\):

\[ b = \frac{ae}{4} \]

Now, substitute this into the eccentricity formula:

\[ \left(\frac{ae}{4}\right)^2 = a^2 (1 - e^2) \] \[ \frac{a^2 e^2}{16} = a^2 (1 - e^2) \]

Step 4: Solve the equation for the eccentricity \( e \).

Since \( a > 0 \), we can divide both sides of the equation by \( a^2 \):

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

Now, we solve for \( e^2 \):

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

Final Computation & Result

To find the eccentricity \( e \), we take the positive square root of the result, as eccentricity must be a positive value for an ellipse (\( 0 < e < 1 \)).

\[ e = \sqrt{\frac{16}{17}} \] \[ e = \frac{4}{\sqrt{17}} \]

The eccentricity of the ellipse is \( \frac{4}{\sqrt{17}} \).