Verified Solution • 2025–26 Session moderate

QJEEM25Macee2: JEE Main 2025 Mathematics - Class 11 JEEM25Macee2 Solution: A circle C of radius 2 lies in the second quadrant and touches…

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)(2,5) and intersects the circle CC at exactly two points. If the set of all possible values of r is the interval (α,β)(\alpha, \beta) , then 3β−2α3 \beta-2 \alpha is equal to :

Multiple Choice Options

  • A 10
  • B 12
  • C 14
  • D 15

Step-by-Step Verified Solution

JEE Main 2025 (Online) 22nd January Morning Shift Mathematics - Circle Question 4 English Explanation

S1:(x+2)2+(y−2)2=22S_1:(x+2)^2+(y-2)^2=2^2

S2:(x−2)2+(y−5)2=r2\mathrm{S}_2:(\mathrm{x}-2)^2+(\mathrm{y}-5)^2=\mathrm{r}^2

Both circle intersect at two points

amp;∴∣r1−r2∣lt;ccc2lt;r1+r2amp;∣r−2∣lt;5lt;2+ramp;⇒3lt;rlt;7amp;r∈(3,7)amp;α=3,β=7amp;3β−2α=15\begin{aligned} & \therefore\left|\mathrm{r}_1-\mathrm{r}_2\right|<\mathrm{c}_{\mathrm{c}} \mathrm{c}_2<\mathrm{r}_1+\mathrm{r}_2 \\ & |\mathrm{r}-2|<5<2+\mathrm{r} \\ & \Rightarrow 3<\mathrm{r}<7 \\ & \mathrm{r} \in(3,7) \\ & \alpha=3, \beta=7 \\ & 3 \beta-2 \alpha=15 \end{aligned}

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring