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 \(a^2\) and \(b^2\)
The equation of the ellipse is in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). From the given equation:
- \(a^2 = 36\)
- \(b^2 = 49\)

### Step 2: Calculate \(a\) and \(b\)
Now, we can find \(a\) and \(b\) by taking the square roots:
- \(a = \sqrt{36} = 6\)
- \(b = \sqrt{49} = 7\)

### Step 3: Determine the relationship between \(a\) and \(b\)
Since \(b > a\) (7 > 6), this indicates that the major axis is along the y-axis.

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

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

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