Multiple choice

A circle touches the $y-$axis at the point $(0,4)$ and cuts the $x-$axis in a chord of length $6$ units. The radius of the circle is

  1. $3$
  2. $4$
  3. $5$
  4. $6$
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C Correct answer
Explanation

Center is (r, 4). Circle touches y-axis at (0,4), so radius is r. Chord on x-axis has length 6, so distance from center to x-axis is sqrt(r^2 - 3^2). Since center is (r, 4), distance is 4. So 4 = sqrt(r^2 - 9). 16 = r^2 - 9, r^2 = 25, r = 5.

AI explanation

Because the circle touches the y-axis at (0, 4), its center is at (r, 4) where r is the radius. The circle cuts the x-axis at points that form a chord of length 6 units, meaning the x-coordinates of these points are (r - 3) and (r + 3), with their y-coordinate being 0. Applying the distance formula between the center (r, 4) and either x-intercept gives r^2 = (r - (r - 3))^2 + 4^2. Solving the equation r^2 = 3^2 + 4^2 yields r^2 = 25, so the radius r is 5.

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