JEE Advanced 2022 Paper 1, Mathematics Section 1 Q8: Involving Two or More Circles

JEE Advanced2022Paper 1Mathematics Section 1

Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _____________.

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Solution

Place $A$ at the origin with $B = (1, 0)$ on the $x$ -axis and $C = (0, 3)$ on the $y$ -axis. Since $∠ A = \frac{π}{2}$ , the hypotenuse $B C$ is a diameter of the circumcircle.

The circumcentre is the midpoint of $B C$ : $(\frac{1}{2}, \frac{3}{2})$ , and the circumradius is $R = \frac{1}{2} 1^{2} + 3^{2} = \frac{10}{2}$ .

A circle tangent to both axes lying in the first quadrant has centre $(r, r)$ and radius $r$ . For internal tangency with the circumcircle, the distance between centres equals $R - r$ :

$(\frac{1}{2} - r)^{2} + (\frac{3}{2} - r)^{2} = \frac{10}{2} - r .$

Squaring: $\frac{1}{4} + \frac{9}{4} - 4 r + 2 r^{2} = \frac{10}{4} - 10 r + r^{2}$ , which simplifies to $r^{2} - 4 r + 10 r = 0$ , i.e., $r (r - 4 + 10) = 0$ .

Since $r > 0$ , $r = 4 - 10 \approx 0.8377 \approx 0.84$ (rounded to two decimals).

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JEE Advanced 2022 Paper 1, Mathematics Section 1 Q8: Involving Two or More Circles

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