QJEEM21Maf1f6: JEE Main 2021 Mathematics - Class 11 JEEM21Maf1f6 Solution: If the variable line 3x + 4y = α lies between the two circles…

Both centers should lie on either side of the line as well as line can be tangent to circle.

(3 + 4 $$-$$ $$\alpha$$) . (27 + 4 $$-$$ $$\alpha$$) < 0

(7 $$-$$ $$\alpha$$) . (31 $$-$$ $$\alpha$$) < 0 $$\Rightarrow$$ $$\alpha$$ $$\in$$ (7, 31) ....... (1)

d₁ = distance of (1, 1) from line

d₂ = distance of (9, 1) from line

$$d_1\ge r_1\Rightarrow \frac{|7-\alpha |}{5}\ge 1\Rightarrow \alpha \in (-\mathrm{\infty },2]\cup [12,\mathrm{\infty })$$ .... (2)

$$d_2\ge r_2\Rightarrow \frac{|31-\alpha |}{5}\ge 2\Rightarrow \alpha \in (-\mathrm{\infty },21]\cup [41,\mathrm{\infty })$$ ....(3)

(1) $$\cap$$ (2) $$\cap$$ (3) $$\Rightarrow$$ $$\alpha$$ $$\in$$ [12, 21]

Sum of integers = 165