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JEE Main 2025 · Mathematics · Straight Lines and Circles · hard

Two circles in the first quadrant each touch both coordinate axes and both pass through the point (1,2). The length of their common chord is

Single correct option

Two circles in the first quadrant each touch both coordinate axes and both pass through the point (1,2)(1,2). The length of their common chord is
  1. A. 2\sqrt2
  2. B. 424\sqrt2
  3. C. 222\sqrt2
  4. D. 22
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Answer: (A) 2\sqrt2

A circle touching both axes in the first quadrant has centre (r,r)(r,r) and radius rr. Passing through (1,2)(1,2) requires (1−r)2+(2−r)2=r2(1-r)^2+(2-r)^2=r^2, i.e. r2−6r+5=0r^2-6r+5=0, so r=1r=1 or r=5r=5: the centres are (1,1)(1,1) and (5,5)(5,5). The common chord is perpendicular to the line of centres y=xy=x, so the second intersection point is the reflection of (1,2)(1,2) in y=xy=x, namely (2,1)(2,1). (Check: (2−1)2+(1−1)2=1(2-1)^2+(1-1)^2=1 and (2−5)2+(1−5)2=25(2-5)^2+(1-5)^2=25.) The chord joins (1,2)(1,2) and (2,1)(2,1), and its length is 1+1=2\sqrt{1+1}=\sqrt2.
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