Question

O is the centre of two concentric circles. AE is a chord of the outer circle and it intersects the inner circle at points B and D. C is a point on the chord in between B and D. What is the value of AC/CE?

A. BC/CD = 1

B. A third circle intersects the inner circle at B and D and the point C is on the line joining the centres of the third circle and the inner circle.

• A. the question can be answered by using either statement alone.
• B. the question can be answered by one of the statements alone, but cannot be answered by using the other statement alone.
• C. the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
• D. the question cannot be answered even by using both statements together.

Hint:

Both statements locate C at the midpoint of BD; recall where the perpendicular from the centre meets both chords.

Answer 1

The Correct Option is A

Step 1: Key fact:
The perpendicular from O to chord AE bisects both the chord AE of the outer circle and the chord BD of the inner circle at the same foot M. Hence AM = ME and BM = MD.

Step 2: Using A alone:
\(BC/CD=1\) means C is the midpoint of BD, i.e. \(C=M\). Then \(AC=CE\) and \(AC/CE=1\). A alone is sufficient.

Step 3: Using B alone:
The line joining the centres of two intersecting circles is the perpendicular bisector of their common chord BD. C lies on this line and on BD, so C is the midpoint of BD, again \(C=M\) and \(AC/CE=1\). B alone is sufficient.

Final Answer:
The correct option is (1): the question can be answered by using either statement alone.