Mathematics
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.
Answer
Let two circles touch each other at point P and T is a point on common tangent as shown in the figure below:
As tangents drawn from an external point to a circle are equal in length.
From T, TA and TP are tangents to the circle with centre O'.
TA = TP …..(i)
From T, TB and TP are tangents to the circle with centre O.
TB = TP …..(ii)
From (i) and (ii),
TA = TB.
Hence, proved that tangents drawn to two circles from any point on common tangent are equal in length.
Related Questions
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.
In the figure (ii) given below, ABC is an isosceles triangle with AB = AC. If ∠ABC = 50°, find ∠BDC and ∠BEC.
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.
The figure given below shows two circles with centres A, B and a transverse common tangent to these circles meet the straight line AB in C. Prove that :
AP : BQ = PC : CQ.
