Assertion (A): The director circle of `x^(2)+y^(2)=4` is `x^(2)+y^(2)=8`
Reason(R): The angle between the tangents from any point on `x^(2)+y^(2)=8` to `x^(2)+y^(2)=4` is `(pi)/2`
The correct answer is

Options

A. Both A and R are true and R si the correct explanation of A
B. Both A and R are true and R is not the correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to analyze both the assertion (A) and the reason (R) provided in the question.

**Step 1: Understanding the Assertion (A)**
The assertion states that the director circle of the circle given by the equation \(x^2 + y^2 = 4\) is \(x^2 + y^2 = 8\).
- The radius of the circle \(x^2 + y^2 = 4\) is \(r = 2\) (since \(\sqrt{4} = 2\)).
- The radius of the director circle is given by the formula \(\sqrt{2} \times r\).
- Therefore, the radius of the director circle is \(\sqrt{2} \times 2 = 2\sqrt{2}\).
- The equation of the director circle, which is centered at the origin, is \(x^2 + y^2 = (2\sqrt{2})^2 = 8\).
- Thus, the assertion is **correct**.

**Step 2: Understanding the Reason (R)**
The reason states that the angle between the tangents from any point on the director circle \(x^2 + y^2 = 8\) to the original circle \(x^2 + y^2 = 4\) is \(\frac{\pi}{2}\).
- The tangents drawn from a point on the director circle to the original circle are perpendicular to each other.
- This is a property of the director circle, which is defined as the locus of points from which tangents to the original circle are perpendicular.
- Therefore, the reason is also **correct**.

**Step 3: Conclusion**
Since both the assertion and the reason are correct, and the reason explains the assertion, we conclude that both statements are true, and the reason is the correct explanation of the assertion.

### Final Answer:
Both the assertion (A) and the reason (R) are correct, and R is the correct explanation of A.

---