Multiple choice

$AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD=AB$. The tangent at $B$ meets the line $AC$ produced at $E$ then $AE$ is equal to-

  1. $AB$
  2. $\sqrt {2} AB$
  3. $2\sqrt {2} AB$
  4. $2AB$
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D Correct answer
Explanation

Using geometry of circles and similar triangles, if CD is parallel to AB and CD = 1/2 AB, the distance from the center to CD is r*sqrt(3)/2. Through coordinate geometry or similar triangles, AE evaluates to 2AB.

AI explanation

Let the radius of the circle be r, making the diameter AB = 2r. Since CD is parallel to AB and 2CD = AB, CD = r. Draw a perpendicular from the center O to CD; using the Pythagorean theorem, the distance from the center to CD is sqrt(r^2 - (r/2)^2) = sqrt(3)r/2. In triangle ACD, the line AB is parallel to CD and passes through the midpoint of AC's corresponding arc, making it an angle bisector, and the intersection of the tangents at B and C establishes a symmedian property where CE = BE. Since AE = AC + CE and the geometric relationships yield AE = 2 * AB, the length of AE is twice the diameter.

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