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 0 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°, then OP = a√2" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let O be the center of the circle and the radius of the circle be a.
- Let P be a point outside the circle from which two tangents PM and PN are drawn to the circle, where M and N are the points of tangency.

2. **Identifying the Angles**:
- The angle between the two tangents PM and PN is given as 90°.
- By the properties of tangents, we know that OM is perpendicular to PM and ON is perpendicular to PN.

3. **Triangles Involved**:
- We can analyze triangles OMP and ONP.
- In triangle OMP, OM = a (radius), PM is tangent, and OP is the line connecting O and P.

4. **Congruence of Triangles**:
- Since the angle PMO is 90° and OM = ON, and OP is common to both triangles, triangles OMP and ONP are congruent.
- Therefore, the angles ∠OMP and ∠ONP are equal.

5. **Calculating Angles**:
- Since the total angle ∠MPN is 90°, each of the angles ∠OMP and ∠ONP must be 45° (because they are equal and add up to 90°).

6. **Using Trigonometry**:
- In triangle OMP, we can use the sine function:
- sin(45°) = OM / OP
- Since OM = a, we have sin(45°) = a / OP.
- We know that sin(45°) = 1/√2, so:
- 1/√2 = a / OP
- Rearranging gives OP = a√2.

7. **Conclusion**:
- Therefore, the statement "If the angle between two tangents drawn from a point P to a circle of radius a and center O is 90°, then OP = a√2" is true.

### Final Answer:
**True**. The reason is that the relationship derived from the geometry and trigonometry of the situation confirms that OP = a√2 when the angle between the tangents is 90°.