Question

if the angle between two tangents draw from an external point P to a circle of radius a and centre O, is 90degree,then find the length of OP

Answer 1

Hope this may help uh dear!!

Answer 2

The length OP is $a\sqrt{2}$ units.

• Let the two tangents drawn from an external point P to the circle be PQ and PR.

• Given,

              O is center of the circle with radius a.

              OQ = OR = a

              ∠QPR = 90°

• OP is the angle bisector of  ∠QPR

              ⇒ ∠OPQ = ∠OPR = 45°

• We know that, the tangent drawn to a circle is perpendicular to radius.

               ⇒ ∠OQP =  ∠ORP = 90°

• Now, ΔOPQ is a right angled triangle

                 $sin\alpha = \frac{opp}{hyp}\\sin45 = \frac{OQ}{OP} \\\frac{1}{\sqrt{2} }=\frac{a}{OP} \\OP = a\sqrt{2} units$