Multiple choice

Determine the length of the chord common to the circles $\displaystyle x^{2}+y^{2}=64 and x^{2}+y^{2}-16x=0$

  1. $2\sqrt{3}$
  2. $4\sqrt{3}$
  3. $6\sqrt{3}$
  4. $8\sqrt{3}$
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D Correct answer
Explanation

Circle 1: x^2 + y^2 = 64 (center 0,0, r=8). Circle 2: x^2 - 16x + y^2 = 0 -> (x-8)^2 + y^2 = 64 (center 8,0, r=8). The radical axis (common chord) is x^2 + y^2 - 64 - (x^2 - 16x + y^2) = 0, which is 16x - 64 = 0, so x = 4. Substitute x=4 into x^2 + y^2 = 64: 16 + y^2 = 64, y^2 = 48, y = +/- sqrt(48) = +/- 4*sqrt(3). Length = 4*sqrt(3) - (-4*sqrt(3)) = 8*sqrt(3).

AI explanation

Subtracting the equation x^2 + y^2 - 16x = 0 from x^2 + y^2 = 64 yields the equation of the common chord, x = 4. Substituting x = 4 into the first circle gives 16 + y^2 = 64, so y^2 = 48 and y = 4*sqrt(3). The chord is a vertical line segment from (4, -4*sqrt(3)) to (4, 4*sqrt(3)), giving a total length of 8*sqrt(3).

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