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)`.
Directrix of hyperbola xy = 16 are

Options

A. `x + y = 4sqrt(2)`
B. `x = y = 4sqrt(2)`
C. `x + y = 4 `
D. `x + y = -4 `

Answer

To find the directrix of the hyperbola given by the equation \(xy = 16\), we will follow these steps:

### Step 1: Understand the given hyperbola
The equation \(xy = 16\) represents a rectangular hyperbola. For a rectangular hyperbola, the asymptotes are perpendicular, and its standard form can be expressed as \(xy = c^2\).

### Step 2: Identify the value of \(c\)
From the equation \(xy = 16\), we can see that \(c^2 = 16\). Therefore, we can find \(c\):
\[ c = \sqrt{16} = 4 \]

### Step 3: Relate \(c\) to \(a\)
For a rectangular hyperbola, the relationship between \(c\) and \(a\) is given by:
\[ c^2 = \frac{a^2}{2} \]
Substituting \(c^2 = 16\) into the equation gives:
\[ 16 = \frac{a^2}{2} \]
Multiplying both sides by 2:
\[ a^2 = 32 \]
Taking the square root:
\[ a = \sqrt{32} = 4\sqrt{2} \]

### Step 4: Find the equations of the directrices
For a rectangular hyperbola, the equations of the directrices can be expressed as:
\[ x + y = \pm a \]
Substituting \(a = 4\sqrt{2}\):
\[ x + y = \pm 4\sqrt{2} \]
This gives us two equations:
1. \(x + y = 4\sqrt{2}\)
2. \(x + y = -4\sqrt{2}\)

### Step 5: Finalize the answer
The directrices of the hyperbola \(xy = 16\) are:
1. \(x + y = 4\sqrt{2}\)
2. \(x + y = -4\sqrt{2}\)

### Summary
The directrices of the hyperbola \(xy = 16\) are \(x + y = 4\sqrt{2}\) and \(x + y = -4\sqrt{2}\).

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