Multiple choice

If a circle passes through $P(0,1), Q(0, 9)$ and touches the x - axis, then which of the following statement (s) is / are TRUE ?

  1. centres of circles are $(\pm 3, \pm 5)$

  2. Equation of one of their direct common tangent is y = 10

  3. Radii of both the circle is 5.

  4. length of common chord of circle is 8.
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AI explanation

Let the circle's center be (h, k) and its radius be r, so its equation is (x - h)^2 + (y - r)^2 = r^2 because it touches the x-axis. Substituting the points (0, 1) and (0, 9) gives h^2 + (1 - r)^2 = r^2 and h^2 + (9 - r)^2 = r^2, which simplifies to h^2 - 2r + 1 = 0 and h^2 - 18r + 81 = 0. Solving this system yields r = 5 and h^2 = 9, meaning the centers are (3, 5) and (-3, 5). Because the centers must lie at a height equal to the radius for the circle to touch the x-axis, their coordinates are (3, 5) and (-3, 5).

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