Two tangents drawn at the points A and B of a circle centred at O meet at P. If `angleAOB = 120^(@)` then `angleAPB : angleAPO ` is

Options

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

Answer

To solve the problem, we need to find the ratio of angles \( \angle APB \) to \( \angle APO \) given that \( \angle AOB = 120^\circ \).

### Step-by-Step Solution:

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

2. **Identify the Angles**:
- We know that \( \angle AOB = 120^\circ \).
- Since \( OA \) and \( OB \) are radii of the circle, the angles \( \angle OAP \) and \( \angle OBP \) are both \( 90^\circ \) because the radius is perpendicular to the tangent at the point of tangency.

3. **Analyze Quadrilateral \( AOBP \)**:
- The sum of the angles in quadrilateral \( AOBP \) is \( 360^\circ \).
- Thus, we can write:
\[ \angle AOB + \angle OAP + \angle OBP + \angle APB = 360^\circ \]
- Substituting the known values:
\[ 120^\circ + 90^\circ + 90^\circ + \angle APB = 360^\circ \]
- Simplifying this gives:
\[ 120^\circ + 180^\circ + \angle APB = 360^\circ \]
\[ \angle APB = 360^\circ - 300^\circ = 60^\circ \]

4. **Determine \( \angle APO \)**:
- Triangles \( AOP \) and \( BOP \) are congruent because:
- \( OA = OB \) (radii of the circle),
- \( AP = BP \) (tangents from a point outside the circle),
- \( \angle OAP = \angle OBP = 90^\circ \).
- Therefore, the angles \( \angle AOP \) and \( \angle BOP \) are equal.
- Let \( \angle AOP = \angle BOP = \theta \).
- The sum of angles in triangle \( AOB \) gives:
\[ \theta + \theta + 120^\circ = 180^\circ \]
- Thus:
\[ 2\theta = 60^\circ \implies \theta = 30^\circ \]
- Therefore, \( \angle APO = 30^\circ \).

5. **Calculate the Ratio**:
- Now we have:
\[ \angle APB = 60^\circ \quad \text{and} \quad \angle APO = 30^\circ \]
- The ratio \( \angle APB : \angle APO \) is:
\[ \frac{\angle APB}{\angle APO} = \frac{60^\circ}{30^\circ} = 2 \]
- Thus, the ratio \( \angle APB : \angle APO = 2 : 1 \).

### Final Answer:
The ratio \( \angle APB : \angle APO \) is \( 2 : 1 \).