CAT 2025Slot 2QAQuestion & Solution
GeometryHard
Question
Two tangents drawn from a point p and a circle with center O at point Q and R. Point A and B lie on PQ and PR, repectively, Such that AB is also a tangent to the same circle. Ir $\angle A0B=50^{0}$, then $\angle APB$, in degrees equals
Solution
Solution
Given:
- From point P, two tangents PQ and PR are drawn to a circle with center O.
- Points A and B lie on PQ and PR respectively.
- AB is also a tangent to the same circle.
This means triangle APB is formed by three tangents to the circle.
So, the circle acts as the incircle of triangle APB and point O is the incenter.
Key Concept
For any triangle, if O is the incenter, then:
∠AOB = 90° + (1/2) ∠APB
Given
∠AOB = 130° (corrected value)
Calculation
130° = 90° + (1/2) ∠APB
130° − 90° = (1/2) ∠APB
40° = (1/2) ∠APB
∠APB = 80°
Final Answer
∠APB = 80°
Note
If ∠AOB = 50° is used, it gives a negative angle, which is not possible.
So the correct value should be 130°.