JEE Main 2021 · 18 March · morning · Single correct · +4 / −1

Choose the correct statement about two circles whose equations are given below :
x² + y² - 10x - 10y + 41 = 0
x² + y² - 22x - 10y + 137 = 0

  1. (a) circles have same centre
  2. (b) circles have no meeting point
  3. (c) circles have only one meeting point
  4. (d) circles have two meeting points
Answer and solution

Answer (c) circles have only one meeting point

Solution

Maths sir · 3 steps

Find both centres and radii, then compare the distance between centres with the sum of the radii.

  1. Centre and radius. For x² + y² + 2gx + 2fy + c = 0, the centre is (-g, -f) and the radius is √(g² + f² - c).
    CircleCentreRadius
    C₁(5, 5)√(25 + 25 - 41) = 3
    C₂(11, 5)√(121 + 25 - 137) = 3
  2. Compare distances. Both centres have y = 5, so the distance between them is 11 - 5 = 6. The sum of the radii is 3 + 3 = 6.

    C₁C₂ = r₁ + r₂ = 6

  3. So. The circles touch externally at exactly one point, (8, 5), which lies on both circles. The answer is (c).

The idea to keep. Compare d = C₁C₂ with the radii. If d > r₁ + r₂, no meeting point. If d = r₁ + r₂ or d = |r₁ - r₂|, one point. If d lies strictly between them, two points.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 120 times in JEE Main since 2020. This question is from 2021.