wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

In the given figure, PT is a common tangent to the circles touching externally at P and AB is another common tangent touching the circles at A and B. Prove that: [3 MARKS]

(i) T is the mid-point of AB

(ii) APB=90

(iii) If X and Y are centers of the two circles, show that the circle on AB as diameter touches the line XY.

Open in App
Solution

Concept: 1 Mark
Application: 2 Marks

(i) Since the two tangents to a circle from an external point are equal, we have

TA = TP and TB = TP.

TA = TB [Each equal to TP]

Hence, T bisects AB, i.e., T is the mid-point of AB.

(ii) TA=TPTAP=TPA ...(i)

TB=TPTBP=TPB ...(ii)

Adding (i) and (ii),

TAP+TBP=TPA+TPB=APB

In ΔAPB, By angle sum property, we have

TAP+TBP+APB=2APB = 180

2APB=180

APB=90

(iii) Thus, P lies on the semi-circle with AB as diameter.

Hence, the circle on AD as diameter touches the line XY.


flag
Suggest Corrections
thumbs-up
14
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Introduction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon