JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Circle

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 ____.

Answer: 0.83

Solution

Triangle ABC is right-angled at A with AB = 1, AC = 3. The hypotenuse BC has length

1^{2} + 3^{2} = 10

.

The circumcircle has diameter BC =

10

, so radius

r_{1} = \frac{10}{2}

and center

C_{1}

at the midpoint of BC:

(\frac{1}{2}, \frac{3}{2})

(placing A at origin, B at (1,0), C at (0,3)).

The smaller circle touches AB (the x-axis) and AC (the y-axis), so its center is at

(r, r)

for some radius

r

.

For internal tangency with the circumcircle:

C_{1} C_{2} = r_{1} - r

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

Squaring both sides:

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

r^{2} - r + \frac{1}{4} + r^{2} - 3 r + \frac{9}{4} = \frac{10}{4} - r 10 + r^{2}

r^{2} - 4 r + 10 r = 0

r (r - 4 + 10) = 0

Since

r > 0

r = 4 - 10 \approx 4 - 3.162 = 0.838

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Numerical answer