The tangents drawn at the point P and Q of a circle centered at O meet at A. If `anglePOQ = 120^@`, then what is the ratio of `anglePAQ : anglePAO`?

Options

A. `2:3`
B. `4:1`
C. `2:1`
D. `5:2`

Answer

To solve the problem, we need to find the ratio of angles \( \angle PAQ \) to \( \angle PAO \) given that \( \angle POQ = 120^\circ \).

### Step-by-Step Solution:

1. **Draw the Circle and Points**:
- Draw a circle with center \( O \).
- Mark points \( P \) and \( Q \) on the circumference of the circle.
- Draw tangents at points \( P \) and \( Q \) that meet at point \( A \).

2. **Identify Given Angles**:
- We know that \( \angle POQ = 120^\circ \).

3. **Determine Angles at Point O**:
- Since \( OP \) and \( OQ \) are radii of the circle, the angles \( \angle OAP \) and \( \angle OAQ \) are both \( 90^\circ \) because the radius is perpendicular to the tangent at the point of tangency.

4. **Calculate Remaining Angle**:
- The angles around point \( O \) must sum to \( 360^\circ \). Therefore, we can find \( \angle AOP \) and \( \angle AOQ \):
- Since \( \angle POQ = 120^\circ \), the remaining angle \( \angle AOP + \angle AOQ = 360^\circ - 120^\circ = 240^\circ \).
- Since \( \angle AOP \) and \( \angle AOQ \) are equal (as \( OP \) and \( OQ \) are equal), we can denote each as \( x \):
\[ 2x = 240^\circ \implies x = 120^\circ \]
Thus, \( \angle AOP = \angle AOQ = 120^\circ \).

5. **Find Angle PAQ**:
- Now, we can find \( \angle PAQ \):
\[ \angle PAQ = \angle POQ - \angle AOP = 120^\circ - 120^\circ = 0^\circ \]
However, this is incorrect as we should consider the angles formed by the tangents.
- The angle \( \angle PAQ \) is actually half of \( \angle POQ \) because of the properties of tangents:
\[ \angle PAQ = \frac{1}{2} \times \angle POQ = \frac{1}{2} \times 120^\circ = 60^\circ \]

6. **Find Angle PAO**:
- Since \( \angle PAO = 90^\circ \) (as established earlier), we now have:
\[ \angle PAO = 90^\circ \]

7. **Calculate the Ratio**:
- Now we can find the ratio of \( \angle PAQ \) to \( \angle PAO \):
\[ \text{Ratio} = \frac{\angle PAQ}{\angle PAO} = \frac{60^\circ}{90^\circ} = \frac{2}{3} \]

### Final Answer:
The ratio of \( \angle PAQ : \angle PAO = 2 : 3 \).