Class 10 Maths · Circles · NCERT Exercise 9.4 · Question 10

Circles Medium

In Fig. 9.19, the common tangent, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear.

Fig. 9.19

Answer

E O

bisects

∠ A E C

because the tangents

E A

and

E C

to the circle with centre

O

are equally inclined to

E O

; likewise

E O^{'}

bisects

∠ B E D

for the circle with centre

O^{'}

. These angles are vertically opposite, so with

∠ A E C = ∠ B E D = 2 x

∠ O E O^{'} = x + (18 0^{\circ} - 2 x) + x = 18 0^{\circ}

. Hence

E O

and

E O^{'}

are opposite rays of one line, and

O

E

and

O^{'}

are collinear.

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