Multiple choice

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

  1. ${ x }^{ 2 }+{ y }^{ 2 }-4x+2=0$
  2. ${ x }^{ 2 }+{ y }^{ 2 }-4x+1=0$
  3. ${ x }^{ 2 }+{ y }^{ 2 }-8x+8=0$
  4. ${ x }^{ 2 }+{ y }^{ 2 }-4y+2=0$
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A Correct answer
Explanation

Circle center (h, 0), radius r. Touches y=x (x-y=0), so distance from (h,0) to x-y=0 is r: |h-0|/sqrt(2) = r, so h^2 = 2r^2. Chord length 2 on y - x/sqrt(3) = 0. Distance from center to line is sqrt(r^2 - 1^2). Solving leads to x^2 + y^2 - 4x + 2 = 0.

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