Explanation:

Correct option 2: 10
Given:
The length of the chord AB is 10 cm.
Tangents at points A and B intersect outside the circle at P.
OP = 2 x OA
Concept used:
The tangent at any point of a circle is perpendicular to the radius through the point of contact
By characteristics of a kite-shaped figure, the longer diagonal bisects the shorter diagonal.
In a 30-60-90 triangle, the ratio of the sides is always in the ratio of 1:√3: 2.
Calculation:
Let the radius of the circle be R cm.
So, OA = OB = R
Then, OP = 2 x OA = 2R
Tangent segments AP, BP, and radii OA, OB form a kite
So, OP bisects the chord AB.

Then, AQ = BQ = 5 cm
Now, ΔOAP & ΔOBP are two right-angled triangles.
From ΔΟΑΡ,
Cos ∠AOP = OA/OP
⇒ Cos ∠AOP = R/2R
⇒ Cos ∠AOP = 1/2
Cos∠ AOP= Cos 60°
⇒ ∠AOP = 60°
So, ∠OPA = 180° - (90+60)° = 30°
Also, ΔAQO &Δ BQO are two right-angled triangles.
So, ∠OAQ = 180° - (90 +60) = 30°
From ΔAQO,
Cos ∠QAO = AQ/OA
⇒ Cos 30° = 5/OA
⇒ √3/2 = 5/OA
⇒ OA = 10/√3
So, AP = (10/√3) × √3 = 10 cm
: The length of AP is 10 cm.