Q. The circles $x^{2} + y^{2} - 10x + 16 = 0$ and $x^{2} + y^{2} = r^{2}$ intersect each other at two distinct points, if
A) $r < 2$
B) $r > 8$
C) $2 < r < 8$
D) $2 \leq r \leq 8$
Explanation
(c)
The centres and radii of given circles are $C_{1}(5,\ 0),\ C_{2}(0,\ 0)$ and
$r_{1} = \sqrt{25 + 0 - 16} = 3$
$r_{2} = r$
Now, $C_{1}C_{2} = 5$
For intersection of two circle,
$r_{2} - r_{1} < C_{1}C_{2} < r_{1} + r_{2}$
$\Rightarrow \ \ \ r - 3 < 5 < 3 + r$
$\Rightarrow \ \ r < 8$ and $r > 2$
$\Rightarrow \ \ 2 < r < 8$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.