JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Circle

Two circles in the first quadrant of radii

r_{1}

and

r_{2}

touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line

x + y = 2

. Then

r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}

is equal to ______.

Answer: 7

Solution

A circle in the first quadrant touching both coordinate axes has center

(a, a)

and radius

a

, with equation

(x - a)^{2} + (y - a)^{2} = a^{2}

.

The chord intercepted by the line

x + y = 2

has length 2. Using the chord-length formula:

2 a^{2} - d^{2} = 2

where

d

is the perpendicular distance from center to the line.

The distance from

(a, a)

x + y = 2

is:

d = \frac{∣ a + a - 2∣}{2} = \frac{∣2 a - 2∣}{2}

Substituting:

a^{2} - \frac{( 2 a - 2 ) ^{2}}{2} = 1

2 a^{2} - (2 a - 2)^{2} = 2

2 a^{2} - 4 a^{2} + 8 a - 4 = 2

- 2 a^{2} + 8 a - 6 = 0

a^{2} - 4 a + 3 = 0

(a - 1) (a - 3) = 0

r_{1} = 1

and

r_{2} = 3

(or vice versa).

Therefore:

r_{1}^{2} + r_{2}^{2} - r_{1} r_{2} = 1 + 9 - 3 = 7

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Integer answer