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Q.

In figure, the common tangent, AB and CD are tangents to two circles with centers O and O ′   intersect at E.  Is it true that the points O,E, O ′   are collinear?

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a

True

b

False 

answer is A.

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Detailed Solution

Given that the common tangent, AB and CD are tangents to two circles with centers O and O ′   intersect at E.  
We have to find whether the points O,E, O ′   are collinear.
The tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.
The required figure geometry is shown below,
Question ImageOn joining the points OA and OC.
From the figure, ∠AEC=∠DEB   , due to vertically opposite angles. And, OA = OC, due to radii of the same circle. Also, OE = OE, due to the common side.
The tangent drawn at any point of a circle is perpendicular to the radius through the point of contact. So, ∠OAE=∠OCE= 90 °  . So, by using the RHS congruence criterion, ΔOAE≅ΔOCE   .
Therefore, by using CPCT, ∠AEO=∠CEO   .
Similarly, by using CPCT, ∠DE O ′ =∠BE O ′   .
So, ∠AEC=∠DEB   .
Therefore, 1 2 ∠AEC= 1 2 ∠DEB   Thus, ∠AEO=∠CEO=∠BE O ′ =∠DE O ′   .
Hence, all four angles are equal which are bisected by O E  and OE . Thus, O, E ′   and O ′   are collinear.
Hence, it is true that O, E ′   and O ′   are collinear.
The given statement is true.
Therefore, the correct option is 1.
 
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