Statement 1 `:` The locus of the point of intersection of the perpendicular tangents of `x^(2) + y^(2) = 4` is `x^(2) =y^(2) =8` .
Statement 2 `:` The direction circle of `x^(2) + y^(2) = r^(2)` is `x^(2) +y^(2) = ( 2x )^(2)`.

Options

A. Statement -1 is Ture, Statement -2 is True and Statement-2 is a correct explanation for Statement -1
B. Statement -1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement -1
C. Statement-1 is True, Statement-2 is false
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze both statements individually and determine their validity.

### Step 1: Analyze Statement 1
The first statement claims that the locus of the point of intersection of the perpendicular tangents of the circle defined by the equation \(x^2 + y^2 = 4\) is \(x^2 = y^2 = 8\).

1. **Identify the Circle**: The equation \(x^2 + y^2 = 4\) represents a circle centered at the origin (0,0) with a radius of 2.

2. **Perpendicular Tangents**: For a circle, the tangents at any point can be perpendicular to each other. The point of intersection of these tangents will lie on the locus we are trying to find.

3. **Finding the Locus**: The locus of the points of intersection of the perpendicular tangents can be found using the property of the director circle. The radius of the director circle is given by \(r\sqrt{2}\), where \(r\) is the radius of the original circle. Here, \(r = 2\), so the radius of the director circle is:
\[ r\sqrt{2} = 2\sqrt{2} \]

4. **Equation of the Director Circle**: The equation of the director circle is given by:
\[ x^2 + y^2 = (2\sqrt{2})^2 = 8 \]

5. **Conclusion for Statement 1**: Thus, the locus of the point of intersection of the perpendicular tangents is indeed \(x^2 + y^2 = 8\). Therefore, Statement 1 is **True**.

### Step 2: Analyze Statement 2
The second statement claims that the director circle of \(x^2 + y^2 = r^2\) is given by \(x^2 + y^2 = (2x)^2\).

1. **Identify the Director Circle**: The director circle for a circle defined by \(x^2 + y^2 = r^2\) has the equation:
\[ x^2 + y^2 = 2r^2 \]
where \(r\) is the radius of the original circle.

2. **Evaluate the Given Equation**: The statement claims that the director circle is given by \(x^2 + y^2 = (2x)^2\). Expanding this gives:
\[ x^2 + y^2 = 4x^2 \implies y^2 = 3x^2 \]
This is not the correct equation for the director circle.

3. **Conclusion for Statement 2**: Since the correct equation for the director circle is \(x^2 + y^2 = 2r^2\) and not \(x^2 + y^2 = (2x)^2\), Statement 2 is **False**.

### Final Conclusion
- Statement 1 is True.
- Statement 2 is False.

Thus, the correct answer is that Statement 1 is true and Statement 2 is false.