the length of the latusrectum of the ellipse `(x^(2))/(36)+(y^(2))/(49)=1` , is

Options

A. `98//6`
B. `72//7`
C. `72//14`
D. `98//12`

Answer

To find the length of the latus rectum of the ellipse given by the equation \(\frac{x^2}{36} + \frac{y^2}{49} = 1\), we can follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)
The standard form of an ellipse is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, we can identify:
- \(a^2 = 36\) and \(b^2 = 49\).

### Step 2: Calculate \(a\) and \(b\)
To find \(a\) and \(b\), we take the square root of \(a^2\) and \(b^2\):
- \(a = \sqrt{36} = 6\)
- \(b = \sqrt{49} = 7\)

### Step 3: Determine the orientation of the ellipse
Since \(b > a\) (7 > 6), the major axis is vertical.

### Step 4: Use the formula for the length of the latus rectum
For an ellipse where \(b > a\), the length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2a^2}{b} \]

### Step 5: Substitute the values of \(a\) and \(b\)
Now, substituting \(a = 6\) and \(b = 7\) into the formula:
\[ L = \frac{2 \cdot (6^2)}{7} = \frac{2 \cdot 36}{7} = \frac{72}{7} \]

### Step 6: Final answer
Thus, the length of the latus rectum of the ellipse is \(\frac{72}{7}\).

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