JEE MAIN - Mathematics (2021 - 31st August Morning Shift - No. 20)
If the variable line 3x + 4y = $$\alpha$$ 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 $$\alpha$$ is ___________.
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 $$\alpha$$ is ___________.
Answer
165
Explanation
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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
$${d_1} \ge {r_1} \Rightarrow {{|7 - \alpha |} \over 5} \ge 1 \Rightarrow \alpha \in ( - \infty ,2] \cup [12,\infty )$$ .... (2)
$${d_2} \ge {r_2} \Rightarrow {{|31 - \alpha |} \over 5} \ge 2 \Rightarrow \alpha \in ( - \infty ,21] \cup [41,\infty )$$ ....(3)
(1) $$\cap$$ (2) $$\cap$$ (3) $$\Rightarrow$$ $$\alpha$$ $$\in$$ [12, 21]
Sum of integers = 165
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