JEE Main 2025 Jan 22 Shift 1, Mathematics Q14: Involving Two or More Circles

JEE Main2025Jan 22, Shift 1Mathematics

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 $3 β - 2 α$ is equal to :

A
15
B
14
C
12
D
10

Solution

Circle C touches both axes from the second quadrant, so its centre is $(- 2, 2)$ with radius $2$ .

Distance between centres: $d = (2 - (- 2))^{2} + (5 - 2)^{2} = 16 + 9 = 5$ .

Two circles intersect at exactly two points iff $∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$ , i.e. $∣ r - 2∣ < 5 < r + 2$ .

From $5 < r + 2$ : $r > 3$ . From $∣ r - 2∣ < 5$ : $- 3 < r < 7$ . Combined: $3 < r < 7$ .

So $α = 3$ , $β = 7$ , and $3 β - 2 α = 21 - 6 = 15$ .

Concept behind this question

Involving Two or More CirclesNotes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2025 Jan 22 Shift 1, Mathematics Q14: Involving Two or More Circles

Concept behind this question

More previous year questions on Involving Two or More Circles

Practice more Mathematics