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Circle Diameter and Exponential Curves

XAT 2014 · Quantitative Ability & Data Interpretation · Hard · Coordinate Geometry

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Circle C1C_1 has a radius of 3 units. The line segment PQ is the only diameter of the circle which is parallel to the X axis. P and Q are points on curves given by the equations y=axy = a^x and y=2axy = 2a^x respectively, where a<1a < 1. The value of a is:

  1. A.

    1 / 2^(1/6)

  2. B.

    1 / 3^(1/6)

  3. C.

    1 / 6^(1/3)

  4. D.

    1 / 6^(1/2)

  5. E.

    None of the above

Answer

A

Explanation

Since PQ is parallel to X-axis, y-coordinates of P and Q are equal: ax1=2ax2  ⟹  ax1−x2=2a^{x_1} = 2a^{x_2} \implies a^{x_1 - x_2} = 2. Length of diameter PQ=2×3=6  ⟹  x1−x2=−6PQ = 2 \times 3 = 6 \implies x_1 - x_2 = -6 (since a<1a < 1, curve decreases, x1<x2x_1 < x_2). a−6=2  ⟹  a=2−1/6=121/6a^{-6} = 2 \implies a = 2^{-1/6} = \frac{1}{2^{1/6}}.

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