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Distance in Geometry of Circle

XAT 2013 · Quantitative Ability & Data Interpretation · Hard · Coordinate Geometry

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The radius of a circle with centre O is 50\sqrt{50} cm. A and C are two points on the circle, and B is a point inside the circle. The length of AB is 6 cm, and the length of BC is 2 cm. The angle ABC is a right angle. Find the square of the distance OB.

  1. A.

    26

  2. B.

    25

  3. C.

    24

  4. D.

    23

  5. E.

    22

Answer

A

Explanation

AC=62+22=40=210AC = \sqrt{6^2 + 2^2} = \sqrt{40} = 2\sqrt{10}. Let A=(0,0)A = (0,0). C=(6,2)C = (6,2). With radius R=50R = \sqrt{50}, center OO lies on perpendicular bisector of ACAC. Midpoint of AC=(3,1)AC = (3,1). Height of OO above ACAC = 50−10=40\sqrt{50 - 10} = \sqrt{40}. Coordinates of O=(1,7)O = (1, 7) or similar. Distance OB2=26OB^2 = 26.

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