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In figure, common tangents AB and CD to the two circles with centers $O 1$ and $O 2$ intersect at E. Find the relation between AB and CD.

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2 A B = C D

A B = 2 C D

A B = C D

2 A B = 3 C D

answer is A.

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Detailed Solution

Given that common tangents AB and CD to the two circles with centers

O 1

and

O 2

intersect at E.
We have to find the relation between AB and CD.
Question Image

A tangent is a line to a circle that touches the circle at only one point, but a tangent line never crosses the circle.
The tangents segments to a circle from the same external point are congruent. So,

E A = E C … . (1)

For the circle having a center O

1

. Also,

E B = E D … . (2)

For the circle having a center

O 2

.
Therefore, from the equation (1) and (2). So,

E A + E B = E C + E D ⇒ A B = C D

Hence, the relation between AB and CD is

A B = C D

.
Therefore, the correct option is 1.

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