Circles
Two Circles — Range of r for Two Intersections
nta_pyq_2026_jan
Grade 11
Question:
Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4x-2y-4=0$ intersect at two distinct points be the interval $(\alpha,\beta)$. Then $\alpha\beta$ is equal to
Step-by-Step Solution
Key Concept: Circle 1: centre $C_1=(-1,-4)$, radius $r$. Circle 2: $(x-2)^2+(y-1)^2=9$, centre $C_2=(2,1)$, radius $3$. Distance $d=|C_1C_2|=\sqrt{9+25}=\sqrt{34}$.
$\alpha=\sqrt{34}-3$, $\beta=\sqrt{34}+3$. $\alpha\beta=25$.
Correct Answer: 4