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JEE Main 2013

JEE Main 2013 Mathematics question on Circles

The circle passing through (1,−2)(1, -2) and touching the axis of xx at (3,0)(3, 0) also passes through the point
  1. A.
    (−5,2)(-5, 2)
  2. B.
    (2,−5)(2, -5)
  3. C.
    (5,−2)(5, -2)
  4. D.
    (−2,5)(-2, 5)
Show answer and solution

Correct option: C

Solution

Since the circle touches the x-axis at (3, 0), its center must be at (3, k) for some k, and radius = |k|.

The circle passes through (1, -2), so:
(1-3)² + (-2-k)² = k²
4 + 4 + 4k + k² = k²
8 + 4k = 0 → k = -2

Center is (3, -2), radius = 2.

Check (5, -2): distance from (3,-2) = √(4+0) = 2 ✓

The point (5, -2) lies on the circle.

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