Question

In the given figure, PQ and PR are two tangents to a circle with centre O. If $\angle$QPR = 46º, then $\angle$QOR is



(a) 67º (b) 134º (c) 44º (d) 46º [CBSE 2014]

Answer 1

It is given that PQ and PR are two tangents to a circle with centre O.

Now, PQ is a tangent at Q and OQ is the radius through the point of contact Q.

∴ $\angle$OQP = 90º (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

PR is a tangent at R and OR is the radius through the point of contact R.

∴ $\angle$ORP = 90º (Tangent at any point of a circle is perpendicular to the radius through the point of contact)

Also, $\angle$QPR = 46º (Given)

In quadrilateral OQPR,

$\angle$OQP + $\angle$QPR + $\angle$ORP + $\angle$QOR = 360º (Angle sum property)

⇒ 90º + 46º + 90º + $\angle$QOR = 360º

⇒ 226º + $\angle$QOR = 360º

⇒ $\angle$QOR = 360º − 226º = 134º

∴ $\angle$QOR = 134º

Hence, the correct answer is option B.