Find the ecentricity and length of latus rectum of the ellipse `16x^(2) + y^(2) = 16 . `

Answer

To find the eccentricity and the length of the latus rectum of the ellipse given by the equation \(16x^2 + y^2 = 16\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
We start with the equation:
\[ 16x^2 + y^2 = 16 \]
To convert this into standard form, we divide the entire equation by 16:
\[ \frac{16x^2}{16} + \frac{y^2}{16} = \frac{16}{16} \]
This simplifies to:
\[ x^2 + \frac{y^2}{16} = 1 \]
Now, we can express it as:
\[ \frac{x^2}{1^2} + \frac{y^2}{4^2} = 1 \]
From this, we can identify \(a = 1\) and \(b = 4\).

### Step 2: Calculate the eccentricity
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e = \frac{c}{a} \]
where \(c = \sqrt{b^2 - a^2}\).

First, we calculate \(c\):
\[ c = \sqrt{b^2 - a^2} = \sqrt{4^2 - 1^2} = \sqrt{16 - 1} = \sqrt{15} \]
Now, substituting \(c\) and \(a\) into the eccentricity formula:
\[ e = \frac{\sqrt{15}}{1} = \sqrt{15} \]

### Step 3: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b\) and \(a\):
\[ L = \frac{2 \cdot 4^2}{1} = \frac{2 \cdot 16}{1} = 32 \]

### Final Results
Thus, the eccentricity \(e\) is \(\sqrt{15}\) and the length of the latus rectum \(L\) is \(32\).

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