Q6 of 31 Page 233

Prove that the perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.

Let l be a tangent to the circle having centre O. l touches the circle at P. Let m be the perpendicular line to l from P.



We have to prove that m passes through O i.e. O ∈ m.


Proof:


If O m, then we can find such M m that O and M are in the same half plane of l.


T ∈ l is a distinct point from P.


∴ ∠MPT = 90° and ∠OPT = 90°


M and O are points of the same half plane so this is impossible.


Thus, our assumption is wrong.


∴ O ∈ m


∴The perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.


Hence proved.


More from this chapter

All 31 →
4

P is in the exterior of a circle at distance 34 from the centre 0. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.

 5

In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line 1 touches the circle at R and intersects and in B and C respectively. If AB = c, BC = a, CA = b, then prove that

(1) AP + AQ = a + b + c


(2) AB + BR = AC + CR = AP = AQ =


7

Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and B. Prove that and bisects.

 8

and are the tangents drawn to ⨀ (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively. Prove that m∠TPR = 2m∠OTR.