Multiple choice

Consider a family of circles which are passing through $M(1,1)$ and are tangent to x-axis. If $(h,k)$ is the centre of circle, then

  1. $\quad k\ge \cfrac { 1 }{ 2 } $
  2. $-\cfrac { 1 }{ 2 } \le k\le \cfrac { 1 }{ 2 } $
  3. $k\le \cfrac { 1 }{ 2 } $
  4. $0 < k < \cfrac { 1 }{ 2 } $
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A Correct answer
Explanation

The circle passes through (1,1) and is tangent to the x-axis, so its radius is |k|. The equation is (x-h)^2 + (y-k)^2 = k^2. Substituting (1,1): (1-h)^2 + (1-k)^2 = k^2, which simplifies to (1-h)^2 + 1 - 2k + k^2 = k^2, so (1-h)^2 + 1 = 2k. Since (1-h)^2 >= 0, 2k >= 1, so k >= 1/2.

AI explanation

Because the circle is tangent to the x-axis, its radius equals the distance from the center (h,k) to the line y=0, so the radius squared is k squared. The circle passes through M(1,1), so the distance equation gives (1-h) squared plus (1-k) squared equals k squared. Expanding and simplifying this yields h squared minus 2h plus 2k minus 1 equals 0, and solving for k gives k equals (h squared minus 2h plus 1) divided by 2 plus 1/2. Since the first term is a squared real number and is always greater than or equal to 0, k must be greater than or equal to 1/2.

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