JEE Main 2021MathematicsCircleHardNumerical

JEE Main 2021 — Circle Question with Solution

JEE Main 2021 (31 Aug Shift 1)

Question

If the variable line 3x+4y=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

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Show full solutionCorrect answer: 165
Correct answer
165

Step-by-step explanation

Given line 3x + 4y = α
Given circles (x-1)2+(y-1)2=1 & (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
 
⇒ 3 +4 -α 27 + 4 - α <0
 
α ∈ 7, 31 .
 
Line is neither touching nor intersecting that means perpendicular distance from  centre to line must more than or equal to radius.
 
⇒ 3+4-α5 ≥ 1 & 27+4-α5 ≥ 2
 
⇒ α ∈ (-∞, 2] ∪ [12,∞) & α ∈ (-∞, 21] ∪ [41,∞)
 
⇒ α ∈ 12, 21
 
Sum of integer values of α = 12 + 13+ --21=165.

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About this question

This is a previous-year question from JEE Main 2021, covering the Circle chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.