Rectangular hyperbola is the hyperbola whose asymptotes are perpendicular hence its equationis `x^(2) - y^(2) = a^(2)`, if axes are rotated by `45^(@)` in clockwise direction then its equation becomes `xy = c^(2)`.
Length of minor axis of hyperbola `xy = 16` is

Options

A. `4sqrt(2)`
B. 4
C. `8sqrt(2)`
D. 8

Answer

To find the length of the minor axis of the hyperbola given by the equation \( xy = 16 \), we can follow these steps:

### Step 1: Identify the standard form of the hyperbola
The equation \( xy = c^2 \) represents a rectangular hyperbola. In this case, we have \( c^2 = 16 \).

### Step 2: Solve for \( c \)
From the equation \( c^2 = 16 \), we can find \( c \):
\[ c = \sqrt{16} = 4 \]

### Step 3: Use the formula for the length of the minor axis
For a rectangular hyperbola of the form \( xy = c^2 \), the length of the minor axis is given by the formula:
\[ \text{Length of minor axis} = 2\sqrt{2}c \]

### Step 4: Substitute the value of \( c \)
Now, substituting the value of \( c = 4 \) into the formula:
\[ \text{Length of minor axis} = 2\sqrt{2} \cdot 4 \]

### Step 5: Calculate the length of the minor axis
Calculating the above expression:
\[ \text{Length of minor axis} = 8\sqrt{2} \]

### Final Answer
Thus, the length of the minor axis of the hyperbola \( xy = 16 \) is \( 8\sqrt{2} \).

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