For hyperbola xy=4, which of the following is not true ?

Options

A. Equations of transverse axis is `ypmx=0`
B. Eccentricity, `e=sqrt2`
C. Co-ordinates of foci are `(2sqrt(2),2sqrt(2))` and `(-2sqrt(2),-2sqrt(2))` and equation of directrix is given by `x+ypm2sqrt(2)=0`
D. none

Answer

To solve the problem regarding the hyperbola defined by the equation \(xy = 4\), we need to analyze the properties of this hyperbola and determine which statement about it is not true.

### Step-by-step Solution:

1. **Identify the Type of Hyperbola**:
The equation \(xy = 4\) represents a rectangular hyperbola. Rectangular hyperbolas have the general form \(xy = c^2\), where \(c\) is a constant. Here, \(c^2 = 4\), so \(c = 2\).

**Hint**: Remember that rectangular hyperbolas have the form \(xy = c^2\).

2. **Determine the Eccentricity**:
For a rectangular hyperbola, the eccentricity \(e\) is given by the formula \(e = \sqrt{2}\). This is a standard result for rectangular hyperbolas.

**Hint**: The eccentricity of a rectangular hyperbola is always \(\sqrt{2}\).

3. **Find the Foci**:
The coordinates of the foci for the hyperbola \(xy = c^2\) can be found using the formula \((\pm c\sqrt{2}, \pm c\sqrt{2})\). Since \(c = 2\), the foci are at \((\pm 2\sqrt{2}, \pm 2\sqrt{2})\).

**Hint**: Use the formula for foci of a rectangular hyperbola to find their coordinates.

4. **Equation of the Directrix**:
The equations of the directrices for the hyperbola \(xy = c^2\) are given by \(x + y = \pm c\sqrt{2}\). Substituting \(c = 2\), we get the equations \(x + y = \pm 2\sqrt{2}\).

**Hint**: The directrix equations can be derived from the properties of the hyperbola.

5. **Equation of the Transverse Axis**:
For rectangular hyperbolas, the transverse axis is along the line \(y = x\). The length of the transverse axis is not typically defined as it is for standard hyperbolas, but it can be said that the transverse axis is along the line \(y = x\).

**Hint**: The transverse axis for a rectangular hyperbola is aligned with the line \(y = x\).

6. **Evaluate the Given Statements**:
Now, we need to evaluate the provided options regarding the hyperbola \(xy = 4\) to find out which one is not true. We have established the eccentricity, foci, and directrix equations.

- If one of the options states that the eccentricity is something other than \(\sqrt{2}\), or gives incorrect coordinates for the foci or incorrect equations for the directrices, that option would be the one that is not true.

**Hint**: Compare the properties you've derived with the options provided to identify the incorrect statement.

### Conclusion:
After analyzing the properties of the hyperbola \(xy = 4\), we can conclude which statement is not true based on the established facts.