JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Circle

If the variable line

3 x + 4 y = α

lies between the two circles

(x - 1)^{2} + (y - 1)^{2} = 1

and

(x - 9)^{2} + (y - 1)^{2} = 4

, without intercepting a chord on either circle, then the sum of all the integral values of

α

is ______.

Answer: 165

Solution

The circle

(x - 1)^{2} + (y - 1)^{2} = 1

has centre

(1, 1)

and radius

1

; the circle

(x - 9)^{2} + (y - 1)^{2} = 4

has centre

(9, 1)

and radius

2

. For

3 x + 4 y = α

not to cut the first circle,

\frac{∣3 + 4 - α ∣}{5} \geq 1

, i.e.

∣ α - 7∣ \geq 5

, so

α \geq 12

on the side towards the second circle. For the second circle,

\frac{∣27 + 4 - α ∣}{5} \geq 2

, i.e.

∣ α - 31∣ \geq 10

, so

α \leq 21

. Lying between the two circles requires

12 \leq α \leq 21

. The integer values

12, 13, \dots, 21

sum to

\frac{10 ( 12 + 21 )}{2} = 165

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Integer answer

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