Mathematics
A point P is 13 cm from the centre of a circle. The length of the tangent drawn from P to the circle is 12 cm. Find the distance of P from the nearest point of the circle.
Answer
Let T be the point of contact of the tangent from point P to the circle with centre O.
From figure,
OT ⊥ PT (As tangent and radius from point of contact are perpendicular to each other.)
In right-angled triangle OPT
⇒ OP2 = OT2 + PT2
⇒ 132 = OT2 + 122
⇒ 169 - 144 = OT2
⇒ OT2 = 25
⇒ OT = 5 cm.
From figure,
⇒ OA = OT = 5 cm (Radius of the circle.)
⇒ PA = OP - OA = 13 - 5 = 8 cm.
Hence, the distance of P from the nearest point of circle = 8 cm.
Related Questions
In the figure (ii) given below, ABC is an isosceles triangle with AB = AC. If ∠ABC = 50°, find ∠BDC and ∠BEC.
Two circles touch each other internally. Prove that the tangents drawn to the two circles from any point on the common tangent are equal in length.
In the figure (i) given below, ABDC is a cyclic quadrilateral. If AB = CD, prove that AD = BC.
From a point outside a circle, with centre O, tangents PA and PB are drawn. Prove that
(i) ∠AOP = ∠BOP
(ii) OP is the perpendicular bisector of the chord AB.