Write true or false and state reason:
If angle between two tangents drawn from a point `P` to a circle of radius `a` and centre `O` is `90^(@)` then `OP=asqrt2`.

Options

A. true
B. false
C. can not determine
D. none of these

Explanation

To determine whether the statement "If the angle between two tangents drawn from a point P to a circle of radius a and center O is 90 degrees, then OP = a√2" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a circle with center O and radius a.
- From point P, two tangents PT and PR are drawn to the circle, creating an angle ∠TPR of 90 degrees.

2. **Identifying Key Angles**:
- Since PT and PR are tangents to the circle, they are perpendicular to the radius at the point of tangency. Therefore, ∠OTP = 90 degrees, where T is the point of tangency.

3. **Using the Properties of Angles**:
- The angle ∠TPR is given as 90 degrees. By the property of tangents, we know that the angle bisector OP divides this angle into two equal angles. Thus, ∠TPO = ∠RPO = 45 degrees.

4. **Analyzing Triangle PTO**:
- Triangle PTO is a right-angled triangle with:
- ∠TPO = 45 degrees
- OT (the radius) = a
- OP (the distance from point P to center O) is the hypotenuse.

5. **Applying Trigonometric Ratios**:
- In triangle PTO, we can use the sine function:
\[ \sin(45^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{TO}{OP} \]
- We know that \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\) and TO = a (the radius).

6. **Setting Up the Equation**:
- Plugging in the known values:
\[ \frac{1}{\sqrt{2}} = \frac{a}{OP} \]
- Rearranging gives:
\[ OP = a \cdot \sqrt{2} \]

7. **Conclusion**:
- The derived result shows that OP = a√2, which matches the statement given in the question.

### Final Answer:
- The statement is **True**.