Multiple choice

The radius of the circle, which is touched by the line $y=x$ and has its centre on the positive direction of x-axis and also cuts-off a chord of length $2$ units along the line $\sqrt { 3 } y-x=0$, is

  1. $\sqrt { 5 } $
  2. $\sqrt { 3 } $
  3. $\sqrt { 2 } $
  4. $1$
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C Correct answer
Explanation

The center is (h, 0) and the circle touches y=x (x-y=0), so the radius r = |h-0|/sqrt(1^2+(-1)^2) = |h|/sqrt(2). The chord length formula for the line x-sqrt(3)y=0 is 2*sqrt(r^2-d^2)=2, where d is the distance from (h, 0) to the line. Solving these equations yields r = sqrt(2).

AI explanation

Since the centre (r, 0) lies on the x-axis and the circle touches the line y = x, the radius equals the distance from the centre to the line, giving r = r/sqrt(2) or r = 0. To find the non-trivial radius, the circle also cuts a chord of length 2 along the line sqrt(3)*y - x = 0, so the perpendicular distance from the centre to this line is d = r/sqrt(4) = r/2. Applying the chord length formula r^2 = d^2 + 1^2 yields r^2 = (r/2)^2 + 1, which simplifies to 3*r^2/4 = 1, giving the radius as sqrt(2).

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