If the latus rectum of an ellipse is equal to half of the minor axis, then what is its eccentricity ?

Options

A. `(sqrt(3))/(2)`
B. `sqrt(3)`
C. 2
D. `(1)/(sqrt(3))`

Answer

To find the eccentricity of an ellipse given that the latus rectum is equal to half of the minor axis, we can follow these steps:

### Step 1: Understand the definitions
The latus rectum of an ellipse is the line segment perpendicular to the major axis that passes through a focus. The formula for the length of the latus rectum \( L \) of an ellipse is given by:
\[ L = \frac{2b^2}{a} \]
where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

### Step 2: Determine the length of the minor axis
The length of the minor axis is given by:
\[ \text{Minor Axis} = 2b \]
Thus, half of the minor axis is:
\[ \frac{1}{2} \times \text{Minor Axis} = \frac{1}{2} \times 2b = b \]

### Step 3: Set up the equation
According to the problem, the latus rectum is equal to half of the minor axis. Therefore, we can set up the equation:
\[ \frac{2b^2}{a} = b \]

### Step 4: Simplify the equation
To simplify this equation, we can multiply both sides by \( a \) (assuming \( a \neq 0 \)):
\[ 2b^2 = ab \]
Next, we can divide both sides by \( b \) (assuming \( b \neq 0 \)):
\[ 2b = a \]
This gives us the relationship between \( a \) and \( b \):
\[ \frac{b}{a} = \frac{1}{2} \]

### Step 5: Find the eccentricity
The eccentricity \( e \) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Now substituting \( \frac{b}{a} = \frac{1}{2} \):
\[ \frac{b^2}{a^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Thus, we can substitute this into the eccentricity formula:
\[ e = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Final Result
The eccentricity of the ellipse is:
\[ e = \frac{\sqrt{3}}{2} \]

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