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MathematicsNumerical+3Question 8 of 54

JEE Advanced 2022 · Paper 1, Mathematics question 8

Let ABCABC be the triangle with AB=1AB = 1, AC=3AC = 3 and ∠BAC=π2\angle BAC = \dfrac\pi2. If a circle of radius r>0r > 0 touches the sides ABAB, ACAC and also touches internally the circumcircle of the triangle ABCABC, then the value of rr is _____.

Answer and worked solution

Official answer: 0.82

Coordinates. Put A at the origin, B =(1,0)=(1,0) and C =(0,3)=(0,3).

  • The circumcircle has centre M=(12,32)M=(\frac12,\frac32) (midpoint of BC) and radius R=102R=\frac{\sqrt{10}}2.
  • A circle touching both legs has centre (r,r)(r,r).

Internal tangency. The distance between the centres is R−rR-r:
(r−12)2+(r−32)2=(102−r)2.(r-\tfrac12)^2+(r-\tfrac32)^2=\left(\tfrac{\sqrt{10}}2-r\right)^2.
Expanding:
2r2−4r+52=52−10 r+r2  ⟹  r=4−10≈0.84.2r^2-4r+\tfrac52=\tfrac52-\sqrt{10}\,r+r^2\implies r=4-\sqrt{10}\approx0.84.

Topics:Equation of a circleTangents and normalsDistance and section formulae
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