JEE 2025 Mathematics PYQ — A circle C of radius 2 lies in the second quadrant and touches bo… | Mathem Solvex | Mathem Solvex
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JEE 2025 — Mathematics PYQ
JEE | Mathematics | 2025
A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α, β), then 3β – 2α is equal to:
Choose the correct answer:
A.
15
(Correct Answer)
B.
14
C.
12
D.
10
Correct Answer:
15
Explanation
Equation of circle C: (x+2)2+(y−2)2=22 Centre C1(−2,2), r1=2 ∵ Another circle having centre C2(2,5) and radius r intersects the circle C at exactly two points ∴|r_{1}-r_{2}|<C_{1}C_{2}<r_{1}+r_{2} |r-2|<5<r+2 -5<r-2<5 and r+2>5 -3<r<7 and r>3 ∴r∈(3,7) α=3 β=7 3β−2λ=21−6=15
Explanation
Equation of circle C: (x+2)2+(y−2)2=22 Centre C1(−2,2), r1=2 ∵ Another circle having centre C2(2,5) and radius r intersects the circle C at exactly two points ∴|r_{1}-r_{2}|<C_{1}C_{2}<r_{1}+r_{2} |r-2|<5<r+2 -5<r-2<5 and r+2>5 -3<r<7 and r>3 ∴r∈(3,7) α=3 β=7 3β−2λ=21−6=15
Let the point (p,p+1) lie inside the region $E = \{(x…
