Length of a latus rectum of the ellipse `x^2/81+y^2/63=1` is (in units)

Options

A. 9
B. 7
C. 14
D. 18

Answer

To find the length of the latus rectum of the ellipse given by the equation \( \frac{x^2}{81} + \frac{y^2}{63} = 1 \), we can follow these steps:

### Step 1: Identify the values of \( a \) and \( b \)
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From the equation \( \frac{x^2}{81} + \frac{y^2}{63} = 1 \), we can identify:
- \( a^2 = 81 \) which gives \( a = \sqrt{81} = 9 \)
- \( b^2 = 63 \) which gives \( b = \sqrt{63} = 3\sqrt{7} \)

### Step 2: Calculate the length of the latus rectum
The formula for the length of the latus rectum \( L \) of an ellipse is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \( b^2 \) and \( a \):
- \( b^2 = 63 \)
- \( a = 9 \)

Thus, we can calculate:
\[ L = \frac{2 \times 63}{9} \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ L = \frac{126}{9} = 14 \]

### Conclusion
Therefore, the length of the latus rectum of the ellipse is \( 14 \) units.

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