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JEE Main 202131 AugustMorning Shift
Exam
JEE Main
Year
2021
Date
31 August
Shift
Morning Shift
Subject
Mathematics
Chapter
Circle

JEE Main 2021 · 31 August · Morning Shift · Mathematics · Circle

If the variable line 3x + 4y = α lies between the two circles ( x -...

If the variable line 3x+4y=α3x + 4y = \alpha lies between the two circles (x−1)2+(y−1)2=1( x - 1{)}^{2} + ( y - 1{)}^{2} = 1 and (x−9)2+(y−1)2=4,( x - 9{)}^{2} + ( y - 1{)}^{2} = 4, without intercepting a chord on either circle, then the sum of all the integral values of α\alpha is

Correct answer

165

Step-by-step solution

Given line 3x+4y=α3x + 4y = \alpha
Given circles (x−1)2+(y−1)2=1 and (x−9)2+(y−1)2=4( x - 1{)}^{2} + ( y - 1{)}^{2} = 1 \text{ and } ( x - 9{)}^{2} + ( y - 1{)}^{2} = 4
According to the given information centres of both the circles must lie on opposite side of line.
L11.L22<0{L}_{11} . {L}_{22} < 0
⇒(3+4−α)(27+4−α)<0⇒ \left(3 + 4 - \alpha\right)\left(27 + 4 - \alpha\right) < 0
α∈(7,31)\alpha \in \left(7, 31\right).
Line is neither touching nor intersecting that means perpendicular distance from centre to line must more than or equal to radius.
⇒∣3+4−α∣5≥1 and ∣27+4−α∣5≥2⇒ \dfrac{\left|3 + 4 - \alpha\right|}{5} \ge 1 \text{ and } \dfrac{\left|27 + 4 - \alpha\right|}{5} \ge 2
⇒α∈(−∞,2]∪[12,∞) and α∈(−∞,21]∪[41,∞)⇒ \alpha \in ( - \infty , 2 ] \cup [ 12, \infty ) \text{ and } \alpha \in ( - \infty , 21 ] \cup [ 41, \infty )
⇒α∈[12,21]⇒ \alpha \in \left[12, 21\right]
Sum of integer values of α=12+13+−−21\alpha = 12 + 13 + - - 21=165.= 165 .
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