Question

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O is 60 then find the length of OP

Answer 1

We know that tangent is always perpendicular to the radius at the point of contact.

So, ∠OAP = 90

We know that if 2 tangents are drawn from an external point, then they are equally inclined to the line segment joining the centre to that point.

So, ∠OPA = 1/2∠APB
= 1/2×60° = 30°

So, in triangle$OPA$

$\sin \left(\angle OPA\right)=\frac{OA}{OP}$

$\sin {30}^0=\frac{a}{OP}$

Therefore,$OP=2a$.